If Y is a random variable with exponential distribution with mean 1/λ, show that E (Y...

If Y is a random variable with exponential distribution with mean 1/λ, show that E (Y ^ k) = k! / (λ^k), k = 1,2, ...

Expert Solution

here for exponential distribution mgf =M(t)=/(-t)

1st derivative of mgf =M¹(t) =(d/dt)*/(-t) =/(-t)²

2nd derivative of mgf =M²(t) =(d/dt)*/(-t)² =*2!/(-t)³

therefore kth derivative of mgf M^k(t) =(d^k/dt)(/(-t)) =k!* /(-t)^k+1

hence E(Y^k) = M^k(0) =k!* /(-0)^k+1 =k!*/()^k+1 =k!/^k

orchestra answered 3 years ago

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