Let X1 and X2 have independent gamma distributions with parameters α, θ and β, θ, respectively. Let W = X1/(X1 + X2). Use a method similar to that given in the derivation of the F distribution (Example 5.2-4) to show that the pdf of W is
We say that W has a beta distribution with parameters α and β.
Answer to relevant QuestionsLet X have a beta distribution with parameters α and β. (a) Show that the mean and variance of X are, respectively (b) Show that when α > 1 and β > 1, the mode is at x = (α − 1)/(α + β − 2). The lifetime in months of a certain part has a gamma distribution with α = θ = 2. A company buys three such parts and uses one until it fails, replacing it with a second part. When the latter fails, it is replaced by the ...The number of accidents in a period of one week follows a Poisson distribution with mean 2. The numbers of accidents from week to week are independent. What is the probability of exactly seven accidents in a given three ...Let W = X1 + X2 + · · · + Xh, a sum of h mutually independent and identically distributed exponential random variables with mean θ. Show that W has a gamma distribution with parameters α = h and θ, respectively Let X and Y equal the respective numbers of hours a randomly selected child watches movies or cartoons on TV during a certain month. From experience, it is known that E(X) = 30, E(Y) = 50, Var(X) = 52, Var(Y) = 64, and ...
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