# Question: Perform a suitable change of variable to show that the

Perform a suitable change of variable to show that the integral defining the gamma function can be written as

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To illustrate the law of large numbers (see also Exercise 5.54 on page 172), use the normal approximation to the binomial distribution to determine the probabilities that the proportion of heads will be anywhere from 0.49 to ...If X has a geometric distribution with θ = 1/3 , find the formula for the probability distribution of the random variable Y = 4 – 5X. If X has a uniform density with α = 0 and β = 1, show that Y = X– 1/ a with a> 0 has the Pareto distribution of Exercise 6.21 on page 184. With reference to Example 3.12 on page 82, find (a) The probability distribution of U = X + Y; (b) The probability distribution of V = XY; (c) The probability distribution of W = X – Y. Let X1 and X2 be two continuous random variables having the joint probability density Find the joint probability density of Y1 = X21 and Y2 = X1X2.Post your question