36. If the random variable X-GAM(0,a), then what is the nth moment of X about the...
You may use the following facts to answer the questions below Fact 1: Suppose that Xi. . . . , X, are independent and X.* GAM (θ.k.) for -1 -1 Fact 2: If Y GAM(0,n aYGAM(ab,n) for any number a >0 1. Suppose that V-GAM(1m) and let lPa θν, where θ > 0. (a) Show that, for any given positive number a, P> a) is an increasing function of (b) What is the probability distribution of W? (c) Would you...
2.13 Consider the discrete random variable defined by x 0 1 34 5 36 36 36 36 36 36 Compute the mean and the variance.
Suppose X is a positive random variable with density for x > 0 and with moment generating function (a) Use the fact that for x > 0 to prove that (b) Use the result in (a) to find ExpectedValue(x-1) if X ~ Gamma(alpha, beta) where the ExpectedValue(X) = alpha(beta) fr) M, (t) We were unable to transcribe this imageExpectedValue(z-1)= 1 M1(-t)dt
X Frequency 0 110 What is the probability that a random variable X selected at random is more than 2? Is it unusual? 1 36 2 30 3 10 4 8 5 6
5. Find the moment generating function of the continuous random variable X whose a. probability density is given by )-3 or 36 0 elsewhere find the values of μ and σ2. b, Let X have an exponential distribution with a mean of θ = 15 . Compute a. 6. P(10 < X <20); b. P(X>20), c. P(X>30X > 10), the variance and the moment generating function of x. d.
1. A binomial random variable has the moment generating function, (t) E(etx)II1 E(etX) (pet+1-p)". Show that EX] = np and Var(X) = np(1-p) using that EX] = ψ(0) and E(X2] = ψ"(0). 2. Lex X be uniformly distributed over (a,b). Show that E[xt and Var(X) using the first and second moments of this random variable where the pdf of X is f(x). Note that the nth moment of a continuous random variable is defined as EXj-Γοχ"f(x)dx (b-a)2 exp 2
Problem 1 (3 marks): A random variable, X, has u? = 0, the third raw moment is twice the mean and has a skewness of o, what is u?
(9 points) In this question, we will find E(X), the nth moment of X,, where X, N(0,0%) is normally distributed with mean 0 and variance oz. (a) (3 points) A function po() is a probability measure on R if po(t) > 0 V2 € R and Po(x) dx = 1. By showing Lee* dar = Vī, show that po(x) = 2 e is a probability measure. (b) (3 points) Completing the square, compute the moment generating function of X, V20...
Random variable X has MGF(moment generating function) gX(t) = , t < 1. Then for random variable Y = aX, some constant a > 0, what is the MGF for Y ? What is the mean and variance for Y ?
Let σ2 be the variance of a random variable X, show that σ2 = μ′2 − μ2 where μ′2 is the second moment about the origin and μ is the mean of X.