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- Let Q be a continuous random variable with PDFfQ(q)= 6q(1 − q) if 0 ≤ q ≤ 1fQ(q) = 0 otherwiseThis Q represents the probability of success of a Bernoulli random variable X, i.e.,P (X = 1 | Q = q) = q.Find fQ|X (q|x) for x ∈ {0, 1} and all q.Consider two independent random variables X1 andX2 having the same Cauchy distributionf(x) = 1π(1 + x2)for − q < x < qFind the probability density of Y1 = X1 + X2 by usingTheorem 1 to determine the joint probability density ofX1 and Y1 and then integrating out x1. Also, identify thedistribution of Y1.At an uncontrolled T- junction, it is seen that on an average, 2 vehicles turn to the right in each interval of 120 seconds. What is the probability that there will be 3 or 4 vehicles turning to the right? (Hint: The arrival of vehicles was found to be a purely random phenomenon)
- Let X , Y be two random variables with the following joint probability mass function: X/Y 2 6 -2 8 -2 0.109422 0.0270663 0.0223591 0.0600166 3 0.120172 0.0297199 0.0245512 0.0659006 -7 0.139485 0.0344963 0 0.0764918 4 0.130901 0.0323734 0.0552402 0.0717846 a.) Are X and Y independent?Suppose an insect lays a very large number of eggs Y, where Y ~ Po(lambda), and suppose that each egg survives with probability p. Assuming that the egg's survival is independent, on the average, how many eggs will survive?If X is a continuous random variable with X ∼ Uniform([0, 2]), what is E[X^3]?
- Let Y be a discrete random variable. Let c be a constant. PROVE Var (Y) = E (Y2) - E (Y)2Let X1 and X2 be independent chi-square random variables with r1 and r2 degrees of freedom, respectively. Let Y1=(X1/r1)/(X2/r2) and Y2=X2. (a) Find the joint pdf of Y1 and Y2.The United States Department of Agriculture (USDA) found that the proportion of young adults ages 20–39 who regularly skip eating breakfast is 0.2380.238. Suppose that Lance, a nutritionist, surveys the dietary habits of a random sample of size ?=500n=500 of young adults ages 20–39 in the United States. Apply the central limit theorem to find the probability that the number of individuals, ?,X, in Lance's sample who regularly skip breakfast is greater than 126126. You may find table of critical values helpful. Express the result as a decimal precise to three places. Then, Apply the central limit theorem for the binomial distribution to find the probability that the number of individuals in Lance's sample who regularly skip breakfast is less than 9898. Express the result as a decimal precise to three places.
- The United States Department of Agriculture (USDA) found that the proportion of young adults ages 20–39 who regularly skip eating breakfast is 0.2380.238. Suppose that Lance, a nutritionist, surveys the dietary habits of a random sample of size ?=500n=500 of young adults ages 20–39 in the United States. Apply the central limit theorem to find the probability that the number of individuals, ?,X, in Lance's sample who regularly skip breakfast is greater than 126126. You may find table of critical values helpful. Express the result as a decimal precise to three places.Let X1,...,Xn be iid random variables with expected value 0, variance 1, and covariance Cov [Xi,Xj] = ρ, for i≠j. Use Theorem of linearity of expectation to find the expected value and variance of the sum Y = X1 +...+Xn.For a continuous random variable with the pdf function f(x) = 0.09375(4 - x^2) if 2 <= x <= 2, and 0 otherwise, solve the following: (a) Compute P(X > 0). (b) Compute P(−1 < X < 1). (Enter your answer to four decimal places.) (c) Compute P(X < −1.2 or X > 1.2). (Round your answer to four decimal places.) Please show your work, thanks!