PDF Distribution Research Articles

Alternative expressions of the PDF and CDF for Gamma, [formula omitted] and [formula omitted] shadowed distributions with applications in wireless communications

There are two approaches well known in the literature for reducing the complexity in averaged performance analyses of wireless communications systems (i.e., for alleviating the computational difficulty in averaged performance evaluations). The first method, known as the moment generating function MGF-based approach, aims to convert the instantaneous performance measure into an exponential integration or a certain sum of exponential functions of different scales. The second approach, which is the focus of this article, aims to convert the exact probability density function PDF and cumulative distribution function CDF of fading distributions into the sum of exponential functions of various scales. As such, we propose weighted sum of exponential functions as alternative closed-form approximations for the PDF and CDF of gamma distribution, η−μ distribution, and κ−μ shadowed distribution with integer fading parameters, and we present how these proposed expressions can be easily applied to the performance analysis of wireless communication systems operating over Nakagami-m fading channels, η−μ fading channels, and κ−μ shadowed fading channels.

Effect of DC Electric Field on Turbulent Flame Structure and Turbulent Burning Velocity

ABSTRACT Effect of electric field on turbulent premixed flame structure and turbulent burning velocity is investigated through OH-PLIF technique. The ring-plate electrode configuration is used, which the ring electrode is high potential. Flame front structure and turbulent burning velocity are derived to estimate the effect of electric field on turbulent flame. Results show that electric field has similar effects compared with turbulence to some extent in this experiment, which could increase flame volume, turbulent burning velocity and decrease the flame surface density. But there are still substantial distinctions in detail. The perturbation induced by turbulence is the vortex, while that induced by electric field is the directional flow, and it can be verified through the PDF distribution of flame curvature with/without electric field. Effect of electric field is a local effect because it is dependent on the distribution of charged species. The research indicates electric field-assisted combustion is mitigated by the turbulence to some extent, and it may hinder the utilization of electric field at practical application for combustion enhancement, but other aspects need further researches. Two reasons are proposed to explain the mitigation of turbulence. Firstly, the response of turbulent burning velocity to the same perturbation decreases with turbulence intensity. Secondly, the wrinkled structure of turbulent flame mitigates the effect of electric field.

Effect of Ionic Wind Induced by DC Electric Field on Biogas/Air Turbulent Premixed Flame Structure

ABSTRACT Effect of DC electric field on lean premixed biogas turbulent flame structure at low and medium turbulence intensity was studied experimentally using OH-PLIF. OH signal distribution, turbulent burning velocity and flame structure geometric parameters were derived. Results show that downward electric field could broaden the downstream burned gas and push the flame tip down. The flame base is closer to the burner rim in downward electric field for the modification of thermal diffusion and flow modification caused by ionic wind. The effect on flame tip height reduction is more obvious for biogas flame with a higher CO2 ratio. According to the flame structure analysis, the turbulent burning velocity of biogas is increased slightly due to the modification of flow field by ionic wind generated by downward electric field, and the increment decreases with the increase of turbulence intensity. In addition, there’s a decrease in mean flame surface density and an increase in centerline flame brush thickness. The flame surface curvature PDF distribution is nearly not affected by the application of DC electric field. For turbulent flames, the effect of electric field is non-uniform and inconstant, which induces extra flow perturbation. This perturbation is different from turbulence, and the combined effects lead to the variation of flame structure.

The impact of inhomogeneous subgrid clumping on cosmic reionization – II. Modelling stochasticity

ABSTRACT Small-scale density fluctuations can significantly affect reionization, but are typically modelled quite crudely. Unresolved fluctuations in numerical simulations and analytical calculations are included using a gas clumping factor, typically assumed to be independent of the local environment. In Paper I, we presented an improved, local density-dependent model for the sub-grid gas clumping. Here, we extend this using an empirical stochastic model based on the results from high-resolution numerical simulations which fully resolve all relevant fluctuations. Our model reproduces well both the mean density-clumping relation and its scatter. We applied our stochastic model, along with the mean clumping one and the Paper I deterministic model, to create large-volume realizations of the clumping field, and used these in radiative transfer simulations of cosmic reionization. Our results show that the simplistic mean clumping model delays reionization compared to local density-dependent models, despite producing fewer recombinations overall. This is due to the very different spatial distribution of clumping, resulting in much higher photoionization rates in the latter cases. The mean clumping model produces smaller H ii regions throughout most of reionization, but those percolate faster at late times. It also causes a significant delay in the 21-cm fluctuations peak and yields lower non-Gaussianity and many fewer bright pixels in the PDF distribution. The stochastic density-dependent model shows relatively minor differences from the deterministic one, mostly concentrated around overlap, where it significantly suppresses the 21-cm fluctuations, and at the bright tail of the 21-cm PDFs, where it produces noticeably more bright pixels.

Algorithm for extracting intersections in PPV space of filaments

A filament is an important structure for studying star formation, especially intersections of filaments which are believed to be more dense than other regions. Identifying filament intersections is the first step in studying them. Current methods can only extract two-dimensional intersections without considering the velocity dimension. In this paper, we propose a method to identify three-dimensional (3D) intersections by combining Harris Corner Detection and Hough Line Transform, which achieve a precision of 98%. We apply this method for extracting intersection structures of the OMC-2/3 molecular cloud and to study its physical properties and obtain the associated PDF distribution. Results show denser gas is concentrated in those 3D intersections.

Network topology of turbulent premixed Bunsen flame at elevated pressure and turbulence intensity

For turbulent premixed flame surface, folded regions will be formed mainly depending on the turbulence-flame interaction and are significant structures which can increase flame surface area at large scale and enhance the local displacement speed. The conventional methods, such as PDF distribution of curvature, may not properly label the folded regions of turbulent flame because these regions intrude deeply into the products, resulting in a small curvature probability. This problem may be aggravated at elevated pressure and turbulence intensity conditions when more folded regions appear at the turbulent flame front. To identify the folded regions, Network topology of turbulent premixed flame front is constructed using the “visibility” method. Results show that this method can convert the spatial signal of turbulent flame to network topology, labeling the folded regions. Compared with curvature PDF, node degree distribution of network can reflect the mechanism of turbulence-flame interaction when the non-dimensional turbulence intensity is increased by different ways. The network structure of turbulent flame will transfer from sparse to condense when the dimensionless turbulence intensity is increased by pressure and perforated plate, as these two methods will extend the turbulence-flame interaction time and promote the interaction intensity, respectively. However, although the dimensionless turbulence intensity will increase with the augment of outlet velocity, the node degree distribution of network structure of turbulent flame front keeps almost constant. This is caused by the reduced turbulence-flame interaction time. It suggests that the turbulence-flame interaction time is an factor as important as the dimensionless turbulence intensity in turbulent premixed combustion. For forced-turbulent premixed flame at elevated turbulence intensity, the “bending phenomenon” will be hidden if the outlet velocity is not taken into account, as the outlet velocity is related to the turbulence-flame interaction time.

Separate estimation of laser micro-Doppler parameters based on maximum likelihood schemes

Laser micro-Doppler (MD) effect is capable of obtaining obvious modulation in weak vibration detection. It helps to estimate target micro-motion parameters with high precision, which may extend the application field of MD to subtle identification and recognition. In laser detection, the multiple scattering points in the field of view will generate the single-channel multi-component (SCMC) signal. Moreover, the micro-Doppler features of each component will be overlapped in the time-frequency domain because of the similar micro-motion parameters. The overlapped SCMC signal makes the estimation of the MD parameters a very difficult problem, and there has been no good method so far. In this paper, a separate parameter estimator based on the maximum likelihood framework is proposed to deal with this underdetermined problem. First, the detailed period scanning method is presented to improve the estimation accuracy of micro-motion frequency from the singular value ratio (SVR) spectrum. Further, the amplitude ratio information of each component is extracted from the SVR spectrum. Then, the closed-form expressions of the maximum likelihood estimation (MLE) for the remaining micro-motion parameters are derived, where the mean likelihood estimation is used to approximate to the performance of MLE. The high nonlinearity and multi-peak distribution shape of the likelihood function (LF) in laser MD signal will lead to incorrect estimation result. To this end, a new LF based on the energy spectrum characteristics is designed. The new LF acts as a smoothing filter to the probability density function, through which the ideal PDF distribution form that has only one smooth peak is obtained. With this modification, the requirements for the initialization are reduced and the robustness in low SNR situation is increased. The Markov chain Monte Carlo sampling is employed to implement the MLE. The Gibbs method is chosen to solve the multi-dimensional parametric problems, and the detailed process is listed. In the end, the simulation results prove the feasibility and high efficiency of the proposed method. The accuracy of parameter estimation reaches the Cramer-Rao boundary. The inverse Radon transform is used as a comparison with the experiment, and the results show the precise estimation advantage of the presented method.

DNS analysis of incipient drop impact dynamics using an accurate level set method

A series of 2D direct numerical simulations were performed with an accurate level set method for single drop impacts. The adopted ACLS method was validated to be efficient with perfect mass conservation in both normal and oblique impacts. A square-root correction for neck bases was modified in accuracy as well as scope of applications. In addition, process of jet formation and evolution was studied to reveal internal dynamics in drop impacts. It's found that pressure gradient and vortex are coexisting and completive reasons for jet topology while the inclined angle has a significant effect on them. Mechanisms of jet formation and evolution are different in the front and back necks. With the help of PDF distribution and correction calculation, a compromise in the competition is observed. This work lays a solid foundation for further studies of dynamics in gas–liquid flows.

Structural changes in liquid Fe and Fe–B alloy on cooling

The existence of liquid–liquid phase transitions has been suggested for liquid Fe and some of its alloys. In order to shed light on this phenomenon in-situ cooling of the Fe80B20 and Fe melts was performed under a synchrotron beam radiation. It allowed for the continuous acquisition of X-ray diffraction spectra in the supercooled liquid region below the liquidus temperature. The procedure allowed monitoring of the structural changes in the supercooled liquid state using real-space atomic pair PDF(R) and radial RDF(R) distribution functions. The results are discussed in comparison with those of molecular dynamics simulation for liquid Fe.

Spectral Characteristics of Tropical Anvils Obtained by Combining TRMM Precipitation Radar with Visible and Infrared Scanner Data

The spectral characteristics of anvils in tropical areas (25°S–25°N) have been investigated on the basis of data from the tropical rainfall measuring mission’s (TRMM) precipitation radar (PR) and a visible and infrared scanner (VIRS), from 1998 to 2007. The anvils’ vertical structures were captured by TRMM PR and categorized into two subtypes: ice anvils with an echo base of ≥6 km and mixed anvils with an echo base between 3 and 6 km. Visible and infrared signals for the anvils, which are from reflectance at 0.63 and 1.6 μm (hereafter referred to as RF1 and RF2, respectively) and the equivalent brightness temperatures of a black body at 3.7, 10.8, and 12.0 μm (hereafter referred to as TB3, TB4, and TB5, respectively), were derived simultaneously by use of TRMM VIRS. The findings reveal that the normalized frequency distribution (i.e., probability density functions, PDF) of anvil tops and bases have a bimodal distribution whereas that of anvil thickness has a single-peak curve. For visible signals, the PDF distribution of RF1 (RF2) for anvils, mixed anvils, and ice anvils has an approximately symmetric distribution with a tropics-wide averages of 0.74, 072, and 0.80 (0.21, 0.21, and 0.20), respectively. It can be concluded that ice anvils are optically thicker and contain many more ice-cloud droplets at the cloud top than mixed anvils. RF1 of anvils is usually lower over land than over ocean, by ~0.1, whereas RF2 of anvils is usually higher over land than over ocean, by ~0.3. This implies that anvil clouds have thinner optical depth and their cloud tops consist of many more small ice droplets over land than over ocean. For infrared signals, TB4 is regarded as a representative channel. The PDF distribution of TB4 for anvils and mixed anvils is broad, with tropics-wide averages of 229.2 and 232 K, respectively. They contain two peaks and the secondary peak lies at a much lower value. For ice anvils, the PDF distribution of TB4 is a single-peak curve with a tropics-wide average of 219.5 K. Mean TB4 of anvils, mixed anvils, and ice anvils is usually lower over land than over ocean, by ~3.6 K, ~1.6 K, and 5 K, respectively. In addition, land–ocean differences between RF1-to-RF2 reflectance ratios are more obvious than those between TB4 and TB5 brightness temperatures. Furthermore, these spectral signals have been proved to be a potentially useful way of revealing the presence of anvils over land and ocean, and especially for separating anvils into the mixed and ice subtypes. Finally, long-term mean values reveal that the spectral signals of anvils and of stratiform and convective precipitating clouds are indeed different, and that the difference varies with different geographical location. In real time, however, uncertainty still exists when only spectral signals are used to discriminate anvils from stratiform and convective precipitating clouds, which means active satellite observations are indispensable.

Fuzzy Monte Carlo Simulation using point-cloud-based probability–possibility transformation

Fuzzy Monte Carlo Simulation (FMCS) uses both the probability density function ( pdf) and possibility distributions (e.g., fuzzy numbers) to model the uncertainty/imprecision associated with the input parameters and, then, to simulate the uncertainty/imprecision associated with the output parameters. A probability–possibility transformation is needed to transfer the information of a fuzzy number into its equivalent pdf, while performing the simulation. This study deals with an approach of FMCS that uses a point-cloud-based probability–possibility transformation. Let x( t), t = 0,1,…, n, be a set of points that represents some random states of an uncertain/imprecise quantity. The collection of points ( x( t), x( t+ i)), t = 0,…, n– i, i∈ {1,2,…} is called point-cloud, providing a visual/computational representation of variability, modality, and ranges associated with the quantity. This study identifies the pdf and possibility distribution (fuzzy number) underlying a given point-cloud. Using these distributions, the relationships between the triangular fuzzy number and unimodal pdf (normal/uniform distributions) are identified. Two numerical examples are described elucidating the effectiveness of the proposed transformation. The first example deals with the issue of monitoring a FMCS process. The other example deals with the issue of making a decision by using FMCS.

Can 21-cm observations discriminate between high-mass and low-mass galaxies as reionization sources?

The prospect of detecting the first galaxies by observing their impact on the intergalactic medium as they reionized it during the first billion years leads us to ask whether such indirect observations are capable of diagnosing which types of galaxies were most responsible for reionization. We attempt to answer this by considering a set of large-scale radiative transfer simulations of reionization in sufficiently large volumes to make statistically meaningful predictions of observable signatures, while also directly resolving all atomically-cooling halos down to 10^8 M_solar. We focus here on predictions of the 21-cm background, to see if upcoming observations are capable of distinguishing a universe ionized primarily by high-mass halos from one in which both high-mass and low-mass halos are responsible, and to see how these results depend upon the uncertain source efficiencies. We find that 21-cm fluctuation power spectra observed by the first generation EoR/21-cm radio interferometer arrays should be able to distinguish the case of reionization by high-mass halos alone from that by both high- and low-mass halos, together. Some reionization scenarios yield very similar power spectra and rms evolution and thus can only be discriminated by their different mean reionization history and 21-cm PDF distributions. We find that the skewness of the 21-cm PDF distribution smoothed over LOFAR-like window shows a clear feature correlated with the rise of the rms due to patchiness. Measurements of the mean photoionization rates are sensitive to the average density of the regions being studied and therefore could be strongly skewed in certain cases. (abridged)

Bohm-like Dependence of Transport in Scrape-off Layer Plasmas

Nondiffusive radial transport in scrape-off layer (SoL) is investigated using the two-dimensional SoL interchange turbulence simulation code, which reproduces plasma blobs. From nonlinear simulations, it is found that a mesoscale convective cell emerges from interchange turbulence and propagates in the radial direction. Blob structures are formed autonomously and their ballistic propagation dominates SoL radial transport. A high-density tail in the radial profile is reproduced. This indicates the existence of a 2nd SoL. The effective transport coefficient is analyzed and a Bohm-like dependence is found. It is concluded that the flux-driven turbulence in SoL induces Bohm-like “nondiffusive” transport. The PDF of the turbulence-driven flux is also investigated. It is found that the PDF distribution follows a stretched Gaussian distribution.

Technical note: Evaluation of bimodal distribution models to determine meal criterion in heifers fed a high-grain diet

Meals are clusters of feedbunk visit (BV) events that are differentiated from the next meal by a nonfeeding interval that is longer compared with the nonfeeding intervals within a meal. The longest nonfeeding interval considered to be part of a meal is defined as the meal criterion. The objective of this study was to determine which combination of 2 probability density functions [(PDF): Gaussian normal (G), Weibull (W), Log-Normal, Gamma, and Gumbel] used in a bimodal distribution model had the best fit of nonfeeding interval data collected in beef heifers. Feeding behavior traits (572,627 total BV events) were measured in 119 heifers fed a high-grain diet (3.08 Mcal ME/kg DM), using a GrowSafe system for 66 d. The frequency and duration of BV events averaged 75 ± 15 events/d and 73.0 ± 22.3 min/d, respectively. The bimodal PDF combinations were fitted to the log(10)-transformed interval lengths between BV events for each animal, using R mixdist package (2.13). The Akaike Information Criterion (AIC) was used to assess goodness of fit of the 25 bimodal PDF combinations. The PDF model with the least AIC value was selected as the best fit for each individual. A χ(2) analysis of the selected best PDF distribution across individuals revealed that 78.2% of the heifers best fit were G-W or W-W PDF models. The likelihood probability estimates were calculated from the average AIC deviation of each model from the standard G-G model. The G-W likelihood probability estimate was greater (P = 0.001) than the W-W combination (0.997 vs. 0.727). Our analysis indicated the G-W model had a statistically better fit and is most likely the best approach to define meal criterion in beef heifers fed high-grain diets.

A Possible Solution for the Boltzmann Equation

Recently there has been an interest in utilizing recent advances in nonlinear partial differential equations solution methods, the equivalent description by stochastic calculus methods, and the state function level of information entropy to the solution of several nonlinear PDEs that are well known in application to problems of gases and fluids and regular to turbulent flow in physics. A recent reported solution has been to the Boltzmann equation. Previously advances were made in solutions to the Navier-Stokes equations of gaseous flow. In this article we derive a possible solution to the 3D Boltzmann equation utilizing transformation methods at the macroscopic functional level, the PDF distribution and stochastic differential equation level. The collision integrals are evaluated utilizing the hypothesis of semiclassical collisions and are obtained from a maximum entropy approach. The collision integral evaluated beyond the molecular chaos approximation, it is 'added' to the phase space variational principle and the Boltzmann equation is transformed to a Fokker-Planck equation which is solved by several known methods both for computational and mathematical analytical applications.

Atomic structure changes and phase transformation behavior in Pd–Si bulk glass-forming alloy

Synchrotron-radiation studies allowed continuous acquisition of X-ray diffraction spectra for binary Pd 81Si 19 bulk metallic glass forming alloy in the glassy, crystalline and liquid state. Such an investigation allowed determination of structural changes in the liquid and glassy states using real-space radial RDF(R) and atomic pair PDF(R) distribution functions. Crystallization behavior of this alloy was also studied by conventional X-ray diffractometry, transmission electron microscopy and differential scanning calorimetry in isochronal and isothermal mode.

The vaccine paradox

The next decade will likely bring astonishing successes in vaccine biology, discovery, and delivery. Justifiable confidence in this proposition led the Bill & Melinda Gates Foundation last year to pledge US$10 billion to a new Decade of Vaccines. For the world's largest and most influential health foundation, vaccines are the number one priority. The foundation estimates that if vaccine coverage could be scaled up to 90%, the lives of 7·6 million children younger than 5 years could be saved between 2010 and 2019. If a malaria vaccine became available by 2014, this figure could rise by a further 1·1 million. The next decade of vaccines: societal and scientific challengesVaccines against microbial diseases have improved the health of millions of people. In the next decade and beyond, many conceptual and technological scientific advances offer extraordinary opportunities to expand the portfolio of immunisations against viral and bacterial diseases and to pioneer the first vaccines against human parasitic and fungal diseases. Scientists in the public and private sectors are motivated as never before to bring about these innovations in immunisation. Many societal factors threaten to compromise realisation of the public health gains that immunisation can achieve in the next decade and beyond—understanding these factors is imperative. Full-Text PDF Vaccine discovery and translation of new vaccine technologyAn unprecedented increase in new vaccine development has occurred over the past three decades. This activity has resulted in vaccines that protect against an increased range of vaccine-preventable diseases, vaccines that reduce the number of required injections, and vaccines with improved safety and purity. New methods of discovery, such as reverse vaccinology, structural biology, and systems biology, promise new vaccines for different diseases and efficient development pathways for these vaccines. Full-Text PDF Vaccine production, distribution, access, and uptakeFor human vaccines to be available on a global scale, complex production methods, meticulous quality control, and reliable distribution channels are needed to ensure that the products are potent and effective at the point of use. The technologies used to manufacture different types of vaccines can strongly affect vaccine cost, ease of industrial scale-up, stability, and, ultimately, worldwide availability. The complexity of manufacturing is compounded by the need for different formulations in different countries and age-groups. Full-Text PDF The future of immunisation policy, implementation, and financingVaccines have already saved many lives and they have the potential to save many more as increasingly elaborate technologies deliver new and effective vaccines against both infectious diseases—for which there are currently no effective licensed vaccines—such as malaria, tuberculosis, and HIV and non-infectious diseases such as hypertension and diabetes. However, these new vaccines are likely to be more complex and expensive than those that have been used so effectively in the past, and they could have a multifaceted effect on the disease that they are designed to prevent, as has already been seen with pneumococcal conjugate vaccines. Full-Text PDF Addressing the vaccine confidence gapVaccines—often lauded as one of the greatest public health interventions—are losing public confidence. Some vaccine experts have referred to this decline in confidence as a crisis. We discuss some of the characteristics of the changing global environment that are contributing to increased public questioning of vaccines, and outline some of the specific determinants of public trust. Public decision making related to vaccine acceptance is neither driven by scientific nor economic evidence alone, but is also driven by a mix of psychological, sociocultural, and political factors, all of which need to be understood and taken into account by policy and other decision makers. Full-Text PDF

Directionlet-Based Bayesian Filter for SAR Image Despeckling

Synthetic aperture radar (SAR) images are inherently affected by multiplicative speckle noise, which is due to the radar coherent wave. In this paper, we introduce a new SAR despeckling method based on directionlet transform using Bayesian MAP estimator. The directionlet coefficients of the logarithmically transformed reflectance image and the speckle image are modeled Cauchy PDF and additive Gaussian distribution, respectively. Then, we exploit this as a priori information to design a maximum a posterior (MAP) estimator. A new method for estimating the dispersion parameter of Cauchy distribution is also presented. Experiment results, carried out on both synthetic speckled image and real SAR image, show that the proposed scheme removes noise from SAR images more efficiently.

Variations in atomic structural features of a supercooled Pd–Ni–Cu–P glass forming liquid during in situ vitrification

In situ vitrification in a synchrotron beam during cooling of a Pd 42.5Cu 30Ni 7.5P 20 melt allowed continuous acquisition of X-ray diffraction spectra in the supercooled liquid region between the liquidus temperature T l and the glass transition temperature T g for the first time in a metallic liquid alloy. These and subsequent acquisitions on reheating allowed determination of the structural changes in the supercooled and glassy states using the real space atomic pair distribution function PDF(R) and radial distribution function RDF(R). The nearest neighbor (NN) distance in the first coordination shell has a low R peak (pre-peak) that appears and increases on cooling from the liquidus temperature to the glass transition temperature in the supercooled liquid region. In this temperature range reversible expansion of the NN distances in the first coordination shell is observed. The slope of the expansion changes discontinuously at the glass transition. This behavior is explained by the theory of local topological fluctuations.

Comparisons of global cloud ice from MLS, CloudSat, and correlative data sets

Aura Microwave Limb Sounder (MLS) version 2.2 (V2.2) and CloudSat R04 (release 4) ice water content (IWC) and partial‐column ice water path (pIWP) measurements are analyzed and compared to other correlative data sets. The MLS IWC, representing an average over ∼300 × 7 × 4 km3 volume, is retrieved at 215–268 hPa with precision varying between 0.06 and 1 mg/m3. The MLS pIWP products, representing the partial columns over ∼100 × 7 km2 area with the bottom at ∼8, ∼6, and ∼11 km for 115, 240, and 640 GHz, have estimated precisions of 5, 1.5, and 0.8 g/m2, respectively. CloudSat, on the other hand, shows a minimum detectable sensitivity of −31 dBZ in the reflectivity measurement at 94 GHz. CloudSat IWC is an average over ∼1.8 × 1.4 × 0.5 km3 volume, and its precision varies from 0.4 mg/m3 at 8 km to 1.6 mg/m3 at 12 km. The estimated single‐profile precision for CloudSat IWP is ∼9 g/m2. However, these measurements are associated with relatively large systematic error, mostly due to uncertainties in the retrieval assumptions about microphysics, which lead to relatively poor accuracy compared to measurement precision. To characterize systematic differences among various observations and those derived from models, we employ the normalized probability density function (pdf) in the comparisons. CloudSat IWC shows generally consistent slopes of pdf distribution with in situ observations, particularly at ∼12 km where the in situ data come mostly from long‐leg flights. Despite similar IWC morphology found between MLS and CloudSat observations, CloudSat R04 IWC retrieval is higher compared to MLS, especially at 14–17 km where the MLS technique is not limited by sensitivity saturation. The MLS and CloudSat IWC pdf's agree well in the overlapped sensitivity range with relative difference <50%, but the difference appears to increase with IWC. MLS and CloudSat cloud ice measurements are compared with other data sets in terms of monthly map and pdf. Comparisons with European Center for Medium range Weather Forecasting (ECMWF) analyses show that grid box averages of monthly ECMWF IWC are much smaller (by ∼5× and ∼20×) than the same MLS and CloudSat averages. Comparisons of pIWP data from CloudSat and passive sensors reveal large uncertainties associated with passive techniques, such as penetration depth and sensitivity limitation. In particular, retrievals from Moderate Resolution Imaging Spectroradiometer (MODIS) and Advanced Microwave Sounding Unit‐B (AMSU‐B) differ largely in IWP pdf from the CloudSat R04 retrieval, showing CloudSat values generally lower (by ∼5× and ∼8×, respectively) at IWP = 10–500 g/m2 but higher at IWP > 500 g/m2.