a) Suppose that, for-1 ≤ a ≤ 1, the probability density function of (X₁, X₂) is given by f(x₁, x₂) = {1- [[1-a(1-2e-¹)(1-2e-*)]*₁-* ,0 ≤ x1,0 ≤ x₂. otherwise i) Find the marginal distribution of X₁. ii) Find E(X₂X₂).
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- The probability density of the random variable Z isgiven by f(z) = kze−z2for z > 00 for z F 0Find k and draw the graph of this probability density.The random variables X and Y have a joint probability density function given by f(x, y) = way, 0 < x < 3 and 1 < y < x, and 0 otherwise.Sketch the graph of the probability density function over the indicated interval. f(x) = 2x/25, [0, 5] Find the indicated probabilities. a) P(0 < x < 3) b) P(1 < x < 3) c) P(3 < x < 5) d) P(x ≥ 1)
- 1. Let f be the probability density function of a continuous random variable X whose range is (-0, 00), where if a sxsb f(x) ={32 0 otherwise. (a) Explain why a, hence b, must be nonnegative. Graph the function f. (b) Find the value of b'-a. 7 (c) If it is known that P(X 8).2. Suppose that Z, and Z2 have a joint probability density function given by 4 3 (1– 2, z2), 0, f(Z, Z2) = 0 < z1 < 1, 0< z2 < 1 elsewhere a) What is the marginal distribution of Z2.Let X be the random variable having pdf f(x) = 1,0 < x < 1, zero e.w. Compute the probability of the range of the random sample of size 4 (X1, ., X4) is less than 1/2. Hint: Range is R = X(4) – X(1).
- a) Suppose that, for -1 ≤ a ≤ 1, the probability density function of (X₁, X₂) is given by f(x₁, x₂) = {[1- -a(1-2e-*₁) (1 - 2e-x²)]e*1-*2 otherwise ,0 ≤ x₁,0 ≤ x₂. i) Find the marginal distribution of X₁. ii) Find E(X₁X₂). b) Consider a random variable K with parameter p, whose probability mass function (PMF) is given by f(k) pqk-1, k=1,2,. = lo, elsewhere i) Derive the moment generating function of K. ii) Use the result obtained in i), to find the expected value of K. iii) Use the result obtained in i), to find the variance of value of K.The probability density function of a distribution is given by x f(x) = exp(-7). Use differentiation to show that the probability density function has a maximum at x = 0. The moment generating function of a distribution is M(t) = (q + pet)", where p € [0, 1], q = 1 - p and n is a positive integer. Use the moment generating function to find the mean and variance of the distribution in terms of p, q and n.Let X₁, X2, X3, X30 be a random sample of size 30 from a population distributed with the following probability density function: f(x) = (0, -, if 0