Let X1 , X2 , . X7 be a set of seven independent random variables, each
having the exponential distribution with the mean value of 1, 1/2 , 1/3 , ...,1/7 respectively.
(a) Find the expected value and standard deviation of their sum T
≡ X1 + X2 + ... + X7 .
(b) What is the probability that min(X1 , X2 , ..., X7 ) > 0.03?
(c) Find the kurtosis of X1 .
(d) Compute the expected value of X1^3 /(1+X2 ^2) .

(a) E(X1 + X2 + ... + X7) = E(X1) + E(X2) + ..... + E(X7).
Var(X1 + X2 + ... + X7) = Var(X1) + Var(X2) + ..... + Var(X7) since the X's are independent

(b) It is well known that X(1) = min(X1 , X2 , ..., X7 ) has an exponential distribution with parameter (see Exponential distribution - Wikipedia, the free encyclopedia). Use this pdf to calculate Pr(X(1) > 0.33).

(c) What definition/type of kurtosis are you using?

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