3. Let f(x) = x, x > 0 be the pdf of X. (X is the...

 

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3. Let f(x) = x¹, x > 0 be the pdf of X. (X is the sum of two independent exponential random variables with A = 1.) (a) Show that f(r) is a valid pdf. (b) Show that the cdf of X is F(x)=1-(1+r)e for z>0, and show that this satisfies the properties of a cdf. (e) Find the mean of X. (d) Show that the mgf of X is M(t)=(1-1)-² (e) Use the mgf to find the mean and variance of X. 3. Let f(x) = x¹, x > 0 be the pdf of X. (X is the sum of two independent exponential random variables with A = 1.) (a) Show that f(r) is a valid pdf. (b) Show that the cdf of X is F(x)=1-(1+r)e for z>0, and show that this satisfies the properties of a cdf. (e) Find the mean of X. (d) Show that the mgf of X is M(t)=(1-1)-² (e) Use the mgf to find the mean and variance of X.

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