Question

(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

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 )

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).

Let ? and ? be two independent random variables with moment
generating functions ?x(?) = ?t^2+2t and
?Y(?)=?3t^2+t . Determine the
moment generating function of ? = ? + 2?. If possible, state the
distribution name (and include parameter values) of the
distribution of ?.

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

Y is a continuous random variable with a probability
density function f(y)=a+by and 0<y<1. Given E(Y^2)=1/6,
Find:
i) a and b.
ii) the moment generating function of Y. M(t)=?

Let X denote a random variable with probability density
function
a. FInd the moment generating function of X
b If Y = 2^x, find the mean E(Y)
c Show that moments E(X ^n) where n=1,4 is given by:

10pts) Let Y be a continuous random variable with density
function f(y) = 1 2 e −|y| , −∞ < y < ∞ 0, elsewhere (a) Find
the moment-generating function of Y . (b) Use the moment-generating
function you find in (a) to find the V (Y ).

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