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homework_5_2005

homework_5_2005 - Y 2 Y σ = 3 2 dx(x X f 3 x Y m and σ...

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Homework Set #5 Problem 1 X has probability density function as shown below. ) x ( f X 2 = otherwise 0, 1 x 0 , x 2 ) x ( X f 0 1 X Calculate the mean value , variance and second initial moment . Verify the relaton E . X m 2 X σ ] 2 X [ E 2 X 2 X m ] 2 X [ σ + = 3 2 dx (x) X f 2 ) Y m 3 (x 2 Y Problem 2 X has uniform distribution between 2 and 3. Consider a new variable . 3 X Y = (a) Sketch the function . Y(X) (b) Find the probability density function of Y. (c) Calculate the mean value and variance of X. (d) Using the probability density function found in (b), calculate the mean value and variance of Y. (e) Verify that m and can be obtained also as
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Unformatted text preview: Y 2 Y σ ∫ = 3 2 dx (x) X f 3 x Y m and σ . ∫ − = Problem 3 Consider two discrete random variables and , with the joint probability mass function shown in the figure below. (Notice that the distribution is concentrated at four points, with equal probability 0.25 at each point). 1 X 2 X X 2 1 -1 0 -1 1 0.25 0.25 0.25 0.25 X 1 (a) Are and independent? Briefly explain why or why not. 1 X 2 X (b) Find the mean values and , the variances and , and the correlation coefficient ρ between and 1 m 2 m 2 1 σ 2 2 σ 1 X 2 X ....
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