Lesson 20

Section 9 (cont.): Review
November 20th, 2014
Lesson 20
In this lesson, we will examine a few more properties of
transformations of random variables. Mainly we will examine
sums of random variables.
We will also work several examples.
Lesson 20
We list several properties of sums of random variables:
If X1 and X2 are random variables, and Y = X1 + X2 , then
E [Y ] = E [X1 ] + E [X2 ]
and
Var [Y ] = Var [X1 ] + Var [X2 ] + 2Cov [X1 , X2 ].
If X1 , . . . , Xn are random variables, and Y =
then
E [Y ] =
n
X
Pn
i=1
Xi ,
E [Xi ]
i=1
and
Var [Y ] =
n
X
i=1
Var [Xi ] + 2
n X
n
X
i=1 j=i+1
Lesson 20
Cov [Xi , Xj ].
If X1 , . . . , Xn are mutually independent random variables,
then
Var [Y ] =
n
X
Var [Xi ]
i=1
and
MY (t) =
n
Y
MXi (t) = MX1 (t) · MX2 (t) · · · MXn (t).
i=1
If X1 , . . . , Xn and Y1 , . . . , Ym are random variables and
a1 , . . . , an , b, c1 , . . . , cm , d are constants, then
Cov

n
X

ai Xi
i=1
+ b,
m
X

cj Yj + d  =
j=1
n X
m
X
i=1 j=1
Lesson 20
ai cj Cov [Xi , Yj ].
Example (9.8)
Given n independent random variables X1 , . . . , Xn each having
the same variance of σ 2 , and defining
U = 2X1 + X2 + · · · + Xn−1
and
V = X2 + · · · + Xn−1 + 2Xn ,
find the coefficient of correlation between U and V .
Lesson 20
Example (9.9)
Independent random variables X , Y , and Z are identically
distributed. Let W = X + Y . The moment generating
function of W is MW (t) = (.7 + .3e t )6 . Find the moment
generating function of V = X + Y + Z .
Lesson 20
Suppose that X is a random variable with mean µ and
standard variance σ and suppose that X1 , . . . , Xn are n
independent random variables with the same distribution as X .
Let Yn = X1 + · · · + Xn . Then E [Yn ] = nµ and Var [Yn ] = nσ 2 .
As n increases, the distribution of Yn approaches a normal
distribution N(nµ, nσ 2 ). This is called the Central Limit
Theorem. This theorem is the justification for using normal
distributions to approximate the distribution of a sum of
random variables.
Lesson 20
Suppose that X1 , . . . , Xk are independent random variables
P
and Y = ki=1 Xi .
distribution of Xi
binomial B(ni , p)
Poisson λi
geometric p
negative binomial ri , p
normal N(µi , σi2 )
exponential with mean µ
gamma with αi , β
distribution of Y
P
binomial B( ni , p)
P
Poisson λi
negative binomial k, p
P
negative binomial ri , p
P
P
N( µi , σi2 )
gamma with α = k, β = 1/µ
P
gamma with αi , β
Lesson 20
Example (9.10)
Smith estimates his chance of winning a particular hand of
blackjack at a casino is .45, his probability of losing is .5, and
his probability of breaking even on a hand is .05. He is playing
at a $10 table, which means that on each play, he either wins
$10, loses $10 or breaks even, with the stated probabilities.
Smith plays 100 times. What is the approximate probability
that he has won money on the 100 plays of the game in total
(assume separate hands are independent)?
Lesson 20
Example (9.11)
If the number of typographical errors per page typed by a
certain typist follows a Poisson distribution with a mean of λ,
find the probability that the total number of errors in 10
randomly selected pages is 10.
Lesson 20

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