Question

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).

Answer #1

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

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

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)

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.

Let X be a discrete random variable with the range RX = {1, 2,
3, 4}. Let PX(1) = 0.25, PX(2) = 0.125, PX(3) = 0.125.
a) Compute PX(4).
b) Find the CDF of X.
c) Compute the probability that X is greater than 1 but less
than or equal to 3.

Let X be a normal random variance with media 1 and variance 4.
Consider a new variance A random variable T defined below:
T = -1 if X < -2
T = 0 if - 2 ≤ X ≤ 0
T = 1 if x>0
Find the moment generating function of T and, from it, calculate E (T) and Var (T).

Q6/
Let X be a discrete random variable defined by the
following probability function
x
2
3
7
9
f(x)
0.15
0.25
0.35
0.25
Give P(4≤ X < 8)
ــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ
Q7/
Let X be a discrete random variable defined by the following
probability function
x
2
3
7
9
f(x)
0.15
0.25
0.35
0.25
Let F(x) be the CDF of X. Give F(7.5)
ــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ
Q8/
Let X be a discrete random variable defined by the following
probability function :
x
2
6...

Let x be a discrete random variable with the following
probability distribution
x: -1 , 0 , 1, 2
P(x) 0.3 , 0.2 , 0.15 , 0.35
Find the mean and the standard deviation of x

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.

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