What is the moment generating function of the probability density function f(x) = 4x-x^2 for 0<x<2? I know it has something to do with E(e^tx) in other words integrate over 2 to 0 e^tx (4x - x^2) dx, but I don't know how to do it.

How do I use this to find the mean and variance of this distribution? (somebody mentioned Taylor decomposition but I don't know what that is)

Thanks.

Solution Summary

The determination of the Moment Generating Function of a probability density function is discussed in the solution.

3.88
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Use your answer from 3.87 to show that E(Y) = aE(X) +b
3.87: [ If X is a random variable with moment-generatingfunction M(t), and Y is a function of X given by Y=aX+b,
show that the moment-generatingfunction for Y is e^(tb) * M(at) ]
My answer for exercise 3.87 is attached.

** Please see the attached file for the complete solution **
Shirley runs a real estate company. She counts the total number of flats that she sells everyday. Let X_t be the total number of flats she sells on day t (t=1,2,...,7), and let X be the total number of flats she sells for the week. Suppose X_t are independent and iden

1.) Let X be a discrete random variable with probability mass function
Pr {X=k} = c(1+ k^2) for k= -2, -1, 0, 1, 2.
a) Determine c.
b) Determine Pr {X <= 0}
c) Determine the mean of X
d) Why is the previous answer fairly obvious?
e) Determine the variance of X
f) Compute Pr {X=2 | X >= 0}
g) Determine the moment genera

2.5-8. Show that 63/512 is the probability that the fifth head is observed on the tenth independent flip of an unbiased coin.
2.5-9. An excellent free-throw shooter attempts several free throws until she misses.
a) If p= 0.9 is her probability of making a free throw, what is the probability of having the first miss on the 13th

D.) Suppose X1 and X2 are independent exp. random variables, each with mean data and y=X1 and X2. What is the momentgeneratingfunction for the random variable y?
Which choice below is right:
1.) My = (1-t/beta)^-2
2.) My = (1-t/beta)^2
3.) My = (1-beta*t)^-2
4.) My = (1-beta*t)^2
E.) Let X1 and X2 be two di

1) Type i light bulbs function for a random amount of time having mean mui and standard deviation sigmai, i=1,2. A light bulb randomly chosen from a bin of bulbs is a type 1 bulb with probability p, and type 2 with prob. (1-p).
what is the expected value and the variance of the lifetime of this bulb?
2) Mx(t)=exp{2(e^t)-2

Problems on momentgeneratingfunctions
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1. Let X and Y be independent normal rv's, each
with mean mu and variance sigma^2. Use momentgeneratingfunctions to show that X+Y and X-Y
are independent normal rv's.
2. If X and Y are independent and
M_X(t)=exp{2e^t-2} a

Please see the attached file for properly formatted problem descriptions.
(1) The random variable X takes on the values 0, 1,2,3 with respective probabilities
1 12 48 64 125' 125' 125' 125
(a) Find E(X),E(X2) and var (X) (b) Find E((3X ± 2)2 (hint: easy if you use the results in (a))
(2) A contractor's profit on a co