Use the given moment-generating function, Mx(t), to identify the distribution of the random variable, X in...
(3 marks) The moment generating function of a random variable X is given by MX(t) = 24 20 < - In 0.6. Find the mean and standard deviation of X using its moment generating function.
The moment generating function (MGF) for a random variable X is: Mx (t) = E[e'X]. Onc useful property of moment generating functions is that they make it relatively casy to compute weighted sums of independent random variables: Z=aX+BY M26) - Mx(at)My (Bt). (A) Derive the MGF for a Poisson random variable X with parameter 1. (B) Let X be a Poisson random variable with parameter 1, as above, and let y be a Poisson random variable with parameter y. X...
A random variable has a moment generating function given by MX(t) = (e^t + 1)^4/16 . Find the expected value and the variance of the variable Y = 2X + 3
Given f(x) = ( c(x + 1) if 1 < x < 3 0 else as a probability function for a continuous random variable; find a. c. b. The moment generating function MX(t). c. Use MX(t) to find the variance and the standard deviation of X.
10. The moment generating function of the random variable X is given by My(t) = exp{2e* – 2} and that of Y by My(t) = fet +. Assuming that X and Y are independent, find (a) P{X + Y = 2). (b) P{XY = 0}. (c) E(XY).
Suppose that a random variable X has the moment generating function given by M(t) (1- 2t)-1 Find E(X) and V(X)
The moment generating function ф(t) of random variable X is defined for all values of t by et*p(x), if X is discrete e f (x)dx, if X is continus (a) Find the moment generating function of a Binomial random variable X with parameters n (the total number of trials) and p (the probability of success). (b) If X and Y are independent Binomial random variables with parameters (n1 p) and (n2, p), respectively, then what is the distribution of X...
(1 point) Suppose that the moment generating function of a random variable X is My(t) = exp(4e – 4) and that of a random variable Y is My(t) = ( oer + 3)''. If X and Y are independent, find each of the following. (a) P{X + Y = 2} = (b) P{XY = 0} = (c) E[XY] = (d) E[(X+Y)?] =
4. The moment generating function of the normal distribution with parameters μ and σ2 is (t) exp ( μ1+ σ2t2 ) for -oo < t oo. Show that E X)-ψ(0)-μ and Var(X)-ψ"(0)-[ty(0)12-σ2. 5. Suppose that X1, X2, and X3 are independent random variables such that E[X]0 and ElX 1 for i-12,3. Find the value of E[LX? (2X1 X3)2] 6. Suppose that X and Y are random variables such that Var(X)-Var(Y)-2 and Cov(X, Y)- 1. Find the value of Var(3X -...
12. let Mx(1) be the moment generating function of X. Show that (a) Mex+o(t) = eMx(at). (b) TX - Normal(), o?) and moment generating function of X is Mx (0) - to'p. Show that the random variable 2 - Normal(0,1) 13. IX. X X . are mutually independent normal random variables with means t o ... and variances o, o,...,0, then prove that X NOEL ?). 14. If Mx(1) be the moment generating function of X. Show that (a) log(Mx...