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(solution) Find the mean and variance of the random variable x for the following cases: x is a uniformly


Find the mean and variance of the random variable x for the following cases: x is a uniformly distributed random variable, whose pdf is r 1 p x ( x ) = b − a , a ≤ x ≤ b . (P3.3) 0, otherwise Also consider the special case when a = − b . x is a Rayleigh distributed random variable, whose pdf is r x e − x 2 / 2 σ 2 , x > 0 f x ( x ) = o 2 0, otherwise . (P3.4) x is a Laplacian distributed random variable, whose pdf is c f x ( x ) = 2 e − c | x | . (P3.5) y is a discrete random variable, given by y = ), n i =1 x i , where the random vari- ables x i , i = 1, 2, ... , n , are statistically independent and identically distributed (i.i.d.) random variables with the pmf: P ( x i = 1) = p and P ( x i = 0) = 1 − p . Remark y is, in fact, a binomial distributed random variable. However, you do not need the binomial distribution to calculate the mean and variance of y . All you need are the linear property of the expectation operation E {·} and the fact that if U and V are statistically independent random variables, then E { UV }= E { U }· E { V }. Given a sample space n and events such as A , B , C , set operations on these events of union, ( A ∪ B ), intersection ( A ∩ C ), complement ( B ¯ ) can be conveniently visualized geometrically by a drawing called the Venn diagram , shown in Figure 3.21. To use the Venn diagram, the areas of the different events represent the probabilities of these events. Thus the area of n , the whole sample space, is equal to 1. Use Venn diagrams to solve the next set of problems.

 


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