1) Prove that the density for the standard normal distribution Satisfies: f(x) = 1 √2πT ∞ [ f(x)dx = 1 -8
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- 2. Let dt Find the average %3D value of f on the interval [o, 1The 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.Find the maximum likelihood estimator for θ in the pdf f(y; θ) = 2y/(1 − θ^2), θ ≤ y ≤ 1.
- Compute the average value of f(x) Average value = = X 1 4x + 3 on the interval [2,6].For x1, the marginal distribution fx1 (x1) function is validis it a density function)Let X and Y random variables have independent Gamma distributions with X-Gamma(1, 6) and Y-Gamma(2, B). a. Find the joint probability density of Z, = X + Y, Z, = X+Y a. Find the marginal pdf of Z2.
- Suppose that X, Y are independent standard normal RVs. Find the joint pdf of Z, W where Z = XY, W = 3X – 2Y.Compute the average rate of change of f(x) = x2 − 2x on the interval [1, 3]. Find c ∈ (1, 3) whichsatisfies the conclusion of the mean value theorem.Let X have a Cauchy distribution with pdf 1 f(x) = T(1+ x2)' -00SEE MORE QUESTIONS