Exponential Frequency Distribution Research Articles

Modulation of Laser Beams by Atmospheric Turbulence

When laser bcams are propagated through the air, they are modulatcd with a noise-like spectrum <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1,2</sup> having a baseband width the order of hundreds of cycles and a nearly exponential frequency distribution. <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> Hogg <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> used a 2.6-km path; Hinchman and Buck <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> used paths of 9 and 90 miles. In each case the optics alld range were such that the receiver col lected a small fraction of the total beam. Since atmospheric refraction causes twinkling and tearing of the beam, one would expect amplitude modulation of the signal received under these conditions even for constant intensity of the total beam.

An exact test of significance for means of samples drawn from populations with an exponential frequency distribution

The mathematical derivation of a test for determining the fiducial limits of, and significance of difference between, means when the samples are drawn from exponential populations is presented. The test for differences between means takes the particularly simple form of theF test (the ratio of the larger to the smaller mean) with each mean possessed of 2n degrees of freedom,n being the number of cases in the sample. Random sampling, a range of scores upwards from a lower limit of zero, and independence of means from each other are necessary assumptions for the use of the test. Examples of situations in which the test should be used are given, together with a description of the necessary computational procedures. Comparisons of the results of the application of this test with the erroneous application of the critical ratio on actual data show that rather large discrepancies exist between the two tests. Results obtained by applying tests which assume normality to exponential distributions are subject to much error.