Question

a. Show that MmX +n (t) = ent MX (tm), for any constants m and n and the moment generating function for X being MX

b. If X is a geometric random variable with p in (0,1).Compute the moment generating function of X. Determine the μ and σ2 from the moment generating function.

Answer #1

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)

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

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 )

(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

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)

TRUE or FALSE? Do not explain your answer.
(a) If A and B are any independent events, then P(A ∪ B) = P(A)
+ P(B).
(b) Every probability density function is a continuous
function.
(c) Let X ∼ N(0, 1) and Y follow exponential distribution with
parameter λ = 1. If X and Y are independent, then the m.g.f. MXY
(t) = e t 2 /2 1 1−t .
(d) If X and Y have moment generating functions MX and...

Consider a discrete random variable X with probability mass
function P(X = x) = p(x) = C/3^x, x = 2, 3, 4, . . . a. Find the
value of C. b. Find the moment generating function MX(t). c. Use
your answer from a. to find the mean E[X]. d. If Y = 3X + 5, find
the moment generating function MY (t).

X is a random variable with Moment Generating Function M(t) =
exp(3t + t2).
Calculate P[ X > 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.

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