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-generating function M(t), and Y is a function of X given by Y=aX+b,
show that the moment-generating function for Y is e^(tb) * M(at) ]
My answer for exercise 3.87 is attached.

Thank you for your messages, indicating that there is a mistake in the question. Please find the momentgeneratingfunction for x=1, 2, and then please solve the amended question:
Given f(x)=(1/2)^x, for x=0,1,2,..., and zero elsewhere, please show that its momentgeneratingfunction is M(z)=2(2-e^z)^-1. Please use it to calc

Can you please explain me how I can use the momentgeneratingfunctions to find the limiting distribution:
Suppose that Zi for N(0,1) and that Z1, Z2,...are independent. Use momentgeneratingfunctions to find the limiting distribution of... (see attached file).

** 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) 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 expectedvalueand the variance of the lifetime of this bulb?
2) Mx(t)=exp{2(e^t)-2

Given that f(x)=x/2, where 0 < x < 2, please prove that the momentgeneratingfunction is
M(z)=[(2z-1)exp(2z)+1]/2z^2, and please use it to prove that the kth moment about the origin is 2^(k+1)/(k+2) by expanding exponentials.

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

It is attached it is to show that if all Xi are normally distributed then
Y=a1X1+a2X2+...+anXn
Also is normally distributed
But please see the attachment as the question is a bit more specific than that as well. Thanks
File is uploaded in pdf and word