Question

Use the moment generating function Mx(t) to find the mean u and variance o^2. Do not find the infinite series.

Mx(t) = e^[5*((e^t)-1)]

Answer #1

The moment generating function for the random variable X is
MX(t) = (e^t/ (1−t )) if |t| < 1. Find the variance of X.

Let Mx(t) be a moment generating function. Let
Sx (t) = [Mx (t)]2− Mx
(t). Prove that S ′x(0) = µX.

(i) If a discrete random variable X has a moment generating
function
MX(t) = (1/2+(e^-t+e^t)/4)^2, all t
Find the probability mass function of X. (ii) Let X and Y be two
independent continuous random variables with moment generating
functions
MX(t)=1/sqrt(1-t) and MY(t)=1/(1-t)^3/2, t<1
Calculate E(X+Y)^2

Suppose that the moment generating function of a random variable
X is of the form MX (t) = (0.4e^t + 0.6)8 . What is the moment
generating function, MZ(t), of the random variable Z = 2X + 1?
(Hint: think of 2X as the sum two independent random variables).
Find E[X]. Find E[Z ]. Compute E[X] another way - try to recognize
the origin of MX (t) (it is from a well-known distribution)

Find the mean and the variance of the binomial distribution.
(Hint: Use the Moment-Generating Function
to make life easy for you!).

the moment generating function of bernoulli distribution is
=1-p+pe^t. use this to calculate the mean and variance of the
distribution
Please try to explain the solution in words also

Find the moment generating function of each of the following
random variables. Then, use it to find the mean and variance of the
random variable
1. Y, a discrete random variable with P(X = n) = (1-p)p^n, n
>= 0, 0 < p < 1.
2. Z, a discrete random variable with P(Z = -1) = 1/5, P(Z = 0)
= 2/5 and P(Z = 2) = 2/5.

The range of a discrete random variable X is {−1, 0, 1}. Let MX
(t) be the moment generating function of X, and let MX(1) = MX(2) =
0.5. Find the third moment of X, E(X^3).

The range of a discrete random variable X is {−1, 0, 1}. Let
MX(t) be the moment generating function of X, and let MX(1) = MX(2)
= 0.5. Find the third moment of X, E(X^3 )

Question 1: Compute the moment generating
function M(t) for a Poisson random variable.
a) Use M’(t) to compute E(X)
b) Use M’’(t) to compute Var(X)

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