Question

Derive the cdf for an exponential distribution with parameter λ.

Answer #1

Derivation for cdf of exponential distribution:-

Thank you.

Given the exponential distribution f(x) = λe^(−λx), where λ >
0 is a parameter. Derive the moment generating function M(t).
Further, from this mgf, find expressions for E(X) and V ar(X).

Let
X1 and X2 be IID exponential with parameter λ > 0. Determine the
distribution of Y = X1/(X1 + X2).

If X is an exponential random variable with parameter λ,
calculate the cumulative distribution function and the probability
density function of exp(X).

Compute the quantile function of the exponential distribution
with parameter λ. Find its median (the 50th percentile).

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Let X1, X2, . . . , Xn be iid following exponential distribution
with parameter λ whose pdf is f(x|λ) = λ^(−1) exp(− x/λ), x > 0,
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(a) With X(1) = min{X1, . . . , Xn}, find an unbiased estimator
of λ, denoted it by λ(hat).
(b) Use Lehmann-Shceffee to show that ∑ Xi/n is the UMVUE of
λ.
(c) By the definition of completeness of ∑ Xi or other tool(s),
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Show that as n → ∞, Xn/n converges in distribution to an
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