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- The moment generating function can be used to find the mean and variance of the normal distribution. Use derivatives of Mx(t) to verify that E(X) = µ and V(X) = 0².The moment generating function of the exponential distribution with rate A is m(t) = for t < A. Moreover, skewness of a random variable X is defined as X – E(X) E /var(X) Find the skewmess for the standard exponential distribution.1. A continuous random variable X has a probability density function (PDF) p(x) = k( 8-x^2/2) on the interval [0, 4] (a) Find k such that p(x) is a valid PDF. (b) Find P(X ≤ 1). (c) Find the mean, µ, of X. (d) Find the variance, σ 2 , of X.
- 4. The monthly revenue (in millions) of a local gaming company is represented by a continuous random variable X having the probability density f(x) = (1 - (x-1)²), 0 < x < 2, 0, elsewhere Find the mean and variance of X.Find the characteristic function of X² when X has the N(u, o²) distribution.Suppose the random variable T is the length of life of an object (possibly the lifetime of an electrical component or of a subject given a particular treatment). The hazard function hr(t) associated with the random variable T is defined by hr(t) = lims-o- P(t ≤ TLet X represent the number of tires with low air pressure on a randomly chosen car. The probability distribution of X is as follows. 2 3 4 P(x) 0.1 0.2 0.1 0.2 0.4 Send data to ExcelDerive the following:(c) variance(d) moment generating function9. The random variable X has probability density function, f(x) where k 1Let X be a random variable normally distributed with mean μx 1. Find the mean μy of Y. 2. Find the standard deviation oy of Y. 3. Find the PDF fy of Y and evaluate fy (5). (My, oy, fy (5)) = ___________). = 6 and standard deviation ox = 4. Let Y = 4X + 5.Suppose that each individual in a large insurance portfolio incurs losses according to an exponential distribution with mean 1/2,where i varies over the portfolio according to a G(a, 3) mixing distribution. The respective densities of the two distributions are given by Sx (x) = (1/2) exp (-x/à), x > 0, 2 > 0; S(2) =- T(a) 2-l exp(-8i), 2 > 0. Given that the Pareto pdf given by fx (x) = (5 +x)ª+1 * > 0, a > 0, 8> 0. (a) Show that the marginal distribution of losses follows a Pareto distribution, i.e. P(a, 5). (b) Use the mixing formulation of the Pareto to deduce that if X~P(a, 5), then E (X) = .Recommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON