Let Y = Xß, with X Expo(1) and > 0. The distribution of Y is called the Weibull distribution with parameter. This generalizes the Exponential, allowing for non-constant hazard functions. Weibull...

Q: - Let Y = Xß, with X Expo (1) and > 0. The distribution of Y is called the Weibull
distribution with parameter. This generalizes the Exponential, allowing for non-
constant hazard functions. Weibull distributions are widely used in statistics,
engineering, and survival analysis; there is even an 800-page book devoted to this
distribution: The Weibull Distribution: A Handbook by Horst Rinne [23]. For this
problem, let = 3.
(a) Find P(Y >s + t|Y >s) for. Does Y have the memory less property?
(b) Find the mean and variance of Y , and the nth moment E(Yn) for n = 1, 2,... .
(c) Determine whether or not the MGF of Y exists.
Sol: - Now, we are given with which means X has exponential
distribution with parameter . Thus, pdf of X is given by
Now, we are also given with
On taking root on both sides we get
Now, differentiating both sides we get



(


)

Thus, the pdf of Y is given by
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From above results...