Suppose X is a discrete random variable and has the moment generating function 1 3 2 1 Mx(t) = : What is the p.m. f. of X? Find E(X).

Consider random variables Y and X = 7Y + 9 with moment-generating functions 1 e2t – e4t 1- et ebt – emt m, (t) 2 m – b 1– ect Identify the values of the parameters (b, m, c) for X.

Given that the discrete random variable X has the moment generating function 0.2et Mx(t) 1-0.8et find the pmf of X.

6. Suppose the moment generating function of X is M(t) = ² et + e²t+e³t. a) Find the mean of X. b) Find the variance of X. c) Find the pmf of X.

1. Let X have the moment-generating function Mx (t) = (1– 2t )r/2 , t< ½ (a) Find Var ( -5X) + E(X+2X²) (b) Find the moment-generating function of the random variable Z =X+2

Your internal body temperature T in °F is a Gaussian (μ =98.6, σ = 0.4) random variable. In terms of the Φ function, find P[T > 100]. Does this model seem reasonable?

Let X; be random variables and M; (t) be their respective moment generating functions, for i = 1, 2, 3. Solve: Suppose that M₁ (t) = 1 +t+t² + t³+... Is this enough information to determine the type of random variable that X₁ is? Explain why or why not. Determine the type of random variable if there is enough information. Determine if it is possible for M₂ (t) = (-1)" -0 (2n)! possible, determine the type of random variable that X₂. -+2n. Explain why or why not. If it is Suppose that that the expectation of X3 is -1 and its second moment is 1. Determine if this is enough information to find the moment generating function of X3 explicitly. Explain why or why not. If it is enough information, compute it explicitly.

Say what will be the expected value of the momentum (px) of a particle described by the wave function

Let X be an Exponential random variable, with the moment generating function (MGF): M(t)= (1-9t)-¹. (a) Derive the MGF above. (b) Derive E(X) using the MGF. (c) Derive V(X) using the MGF. (d) Find the MGF of Y, where Y = 2X-3.