Using the form of the gamma function of Exercise 6.8, we can write
Change to polar coordinates to evaluate this double integral, and thus show that (12 ) = v p.
Answer to relevant QuestionsIf X has an exponential distribution with the parameter θ, use the distribution function technique to find the probability density of the random variable Y = lnX. Use the transformation-of-variable technique to prove Theorem 6.7 on page 188. Theorem 6.7 If X has a normal distribution with the mean µ and the standard deviation s, then Z = X – µ / σ Has the standard normal ...Consider the random variable X with the probability density (a) Use the result of Example 7.2 to find the probability density of Y = | X|. (b) Find the probability density of Z = X2(=Y2). If X1 and X2 are independent random variables having the geometric distribution with the parameter θ, show that Y = X1 + X2 is a random variable having the negative binomial distribution with the parameters θ and k = 2. Let X and Y be two independent random variables having identical gamma distributions. (a) Find the joint probability density of the random variables U = X / X + Y and V = X + Y. (b) Find and identify the marginal density ...
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