If the probability density of X is given by
Find the probability density of Y = X3. Also, plot the graphs of the probability densities of X and Y and indicate the respective areas under the curves that represent P(1/2 < X < 1) and P(1/8 < Y < 1).
Answer to relevant QuestionsIf X has a uniform density with α = 0 and β = 1, show that the random variable Y = –2 ∙ lnX has a gamma distribution. What are its parameters? If the joint probability distribution of X1 and X2 is given by f(x1, x2) = x1x2 / 36 For x1 = 1, 2, 3 and x2 = 1, 2, 3, find (a) The probability distribution of X1X2; (b) The probability distribution of X1/ X2. If X has the uniform density with the parameters α = 0 and β = 1, use the distribution function technique to find the probability density of the random variable Y = √X. 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 ...Use the result of Exercise 7.45 to show that, if n independent random variables Xi have normal distributions with the means µi and the standard deviations σi, then Y = α1X1 + α2X2 + · · · + anXn has a normal ...
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