Download Summary of Random Variables

MATH 477 { Section E1
7/8/98
Summary of Random Variables
Denition. A random variable is a real-valued function on a sample space.
Denition. The cumulative distribution function (or c.d.f) of the random variable X is dened
for all real numbers b, ,1 < b < 1, by
F (b) = IP fX bg :
A cumulative distribution function F has the following properties:
1. F is a nondecreasing function; that is, if a < b, then F (a) F (b).
2. blim
F (b) = 1.
!1
3. b!,1
lim F (b) = 0.
4. F is right continuous. That is, for any b and any decreasing seqence bn , n 1, that
converges to b, limn!1 F (bn ) = F (b).
Let F be the c.d.f. of random variable X , then
IP fX bg = F (b)
IP fa < X bg = F (b) , F (a)
1
IP fX < bg = nlim
!1 F b , n
Denition. Let X be a discrete random variable. Then, the probability mass function, p(a),
is dened to be
p(a) = IP fX = ag :
The probability mass function p(a) is positive for at most a countable number of values of a.
That is, if X must assume one of the values x1 ; x2; : : :, then
p (xi ) > 0 i = 1; 2; : : : ;
p(x) = 0 all other values of x.
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Summary of Random Variables
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The cumulative distribution function F can be expressed in terms of p(a) by
F (a) =
X
a
allx
p(x) :
Denition. If X is a discrete random variable having a probability mass function p(x), the
expectation or the expected value of X , denoted by IE [X ], is dened by
X
IE [X ] =
x:p(x)>0
x p(x) :
Proposition 5.1. If X is a discrete random variable that takes on one of the values xi, i 1,
with respective probabilities p (xi ), then for any real-valued function g
X
IE [g (X )] = g (xi ) p (xi ) :
i
Corollary 5.1. If a and b are constants, then
IE [aX + b] = aIE [X ] + b :
IE [X n ] =
X
x:p(x)>0
xn p(x).
Denition. If
X is a random variable with mean , then the variance of X , denoted by
Var(X ), is dened by
h
i
Var(X ) = IE (X , )2 :
h i
Var(X ) = IE X 2 , (IE [X ])2.
Var(aX + b) = a Var(X ) .
2
Denition. The square root of the Var(X ) is called the standard deviation of X and we denote
it by SD(X ). That is,
q
SD(X ) = Var(X ) :
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Bernoulli Distribution. If X is distributed as a Bernoulli random variable with parameter p,
then
p(0) = IP fX = 0g = 1 , p
p(1) = IP fX = 1g = p
and
Var(X ) = p(1 , p) :
IE [X ] = p
Binomial Distribution. If X is distributed as a binomial random variable with parameters n
and p, then
p(i) =
and
!
n i
p (1 , p)n,i
i
Var(X ) = np(1 , p) :
IE [X ] = np
Probability mass function for a binomial random variable with parameters n = 10 and p = 0:1:
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
2
4
6
8
10
Probability mass function for a binomial random variable with parameters n = 10 and p = 0:5:
0.5
0.4
0.3
0.2
0.1
0
2
4
6
3
8
10
Summary of Random Variables
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Proposition 7.1. If X is a binomial random variable with parameters (n; p), where 0 < p < 1,
then as k goes from 0 to n, IP fX = kg rst increases monotonically and then decreases
monotonically, reaching its largest value when k is b(n + 1)pc.
Geometric Distribution. If X is distributed as a geometric random variable with parameter
p, then
p(n) = (1 , p)n,1 p
and
Var(X ) = 1 p,2 p :
1
IE [X ] =
p
Probability mass function for a geometric random variable with parameter p = 1=9:
0.2
0.15
0.1
0.05
0
10
20
30
40
50
Negative Binomial Distribution. If X is distributed as a negative binomial random variable
with parameters r and p, then
p(n) =
and
IE [X ] =
r
p
!
n,1 r
p (1 , p)n,r
r,1
Var(X ) = r(1p,2 p) :
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Summary of Random Variables
Probability mass function for a negative binomial random variable with parameters p = 1=9 and
r = 2:
0.08
0.06
0.04
0.02
0
10
20
30
40
50
Poisson Distribution. If X is distributed as a Poisson random variable with parameter ,
then
p(i) =
and
IE [X ] = e, i
i!
Var(X ) = :
A Poisson random variable approximates a binomial random variable with =
large.
Probability mass function for a Poisson random variable with parameter = 0:1:
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
1
2
3
5
4
5
np
and n
Summary of Random Variables
Probability mass function for a Poisson random variable with parameter = 1:
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
2
3
4
5
Probability mass function for a Poisson random variable with parameter = 5:
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
5
10
15
20
Probability mass function for a Poisson random variable with parameter = 10:
0.25
0.2
0.15
0.1
0.05
0
5
10
6
15
20
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