When α and β are integers and 0 < p < 1, we Have
Where n = α + β − 1. Verify this formula when α = 4 and β = 3.
Answer to relevant QuestionsLet W1, W2 be independent, each with a Cauchy distribution. In this exercise we find the pdf of the sample mean, (W1 + W2)/2. (a) Show that the pdf of X1 = (1/2)W1 is (b) Let Y1 = X1 + X2 = W and Y2 = X1, where X2 = (1/2)W2. ...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 ...Let X1 and X2 be independent random variables with respective binomial distributions b(3, 1/2) and b(5, 1/2). Determine (a) P(X1 = 2, X2 = 4). (b) P(X1 + X2 = 7). The number of cracks on a highway averages 0.5 per mile and follows a Poisson distribution. Assuming independence, what is the probability that, in a 40-mile stretch of that highway, there are fewer than 15 cracks? Let T have a t distribution with r degrees of freedom. Show that E(T) = 0 provided that r ≥ 2, and Var(T) = r/(r − 2) provided that r ≥ 3, by first finding E(Z), E(1/√U), E(Z2), and E(1/U).
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