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```Random Variables
A random variable X is a real valued function defined
on the sample space, X: S  R. The set {s  S : X(s) [a,
b] is an event}.
Example 1
Let S be the sample space of an experiment consisting of
tossing two fair dice. Then, S = {(1, 1}, (1, 2),..., (6, 6)}. Let X be
a random variable defined over S that assigns to each
outcome the sum of the dice. Then, X ((1,1)) = 2, X ((1,2)) = 3,
... X ((6,6)) = 12.
Usually, we specify the distribution of a random
variable without reference to the probability space.
If X denotes the random variable that is defined as the sum of two
fair dice, then
If X denotes the random variable that is defined as the sum of two
fair dice, then
P{X = 2} = P{(1, 1)} = 1/36
If X denotes the random variable that is defined as the sum of two
fair dice, then
P{X = 2} = P{(1, 1)} = 1/36
P{X = 3} = P{(1, 2), (2, 1)} = 2/36
If X denotes the random variable that is defined as the sum of two
fair dice, then
P{X = 2} = P{(1, 1)} = 1/36
P{X = 3} = P{(1, 2), (2, 1)} = 2/36
P{X = 4} = P{(1, 3), (3, 1), (2, 2)} = 3/36
If X denotes the random variable that is defined as the sum of two
fair dice, then
P{X = 2} = P{(1, 1)} = 1/36
P{X = 3} = P{(1, 2), (2, 1)} = 2/36
P{X = 4} = P{(1, 3), (3, 1), (2, 2)} = 3/36
P{X = 5} = P{(1, 4), (4, 1), (2, 3), (3, 2)} = 4/36
If X denotes the random variable that is defined as the sum of two
fair dice, then
P{X = 2} = P{(1, 1)} = 1/36
P{X = 3} = P{(1, 2), (2, 1)} = 2/36
P{X = 4} = P{(1, 3), (3, 1), (2, 2)} = 3/36
P{X = 5} = P{(1, 4), (4, 1), (2, 3), (3, 2)} = 4/36
P{X = 6} = P{(1, 5), (5, 1), (2, 4), (4, 2), (3, 3)} = 5/36
P{X = 7} = P{(1, 6), (6, 1), (3, 4), (4, 3), (5, 2), (2, 5)} = 6/36
P{X = 8} = P{(2, 6), (6, 2), (3, 5), (5, 3), (4, 4)} = 5/36
P{X = 9} = P{(3, 6), (6, 3), (5, 4), (4, 5)} = 4/36
P{X = 10} = P{(4, 6), (6, 4), (5, 5)} = 3/36
P{X = 11} = P{(5, 6), (6, 5)} = 2/36
P{X = 12} = P{(6, 6)} = 1/36
The random variable X takes on values X = n, where n = 2,
..., 12. Since the events corresponding to each value are
mutually exclusive, then:
P

12
n2

{ X  n}   n  2 P( X  n)  1.
12
Example 2
Let S be the sample space of an experiment consisting of
tossing two fair coins. Then, S = {(H, H}, (H, T), (T, H), (T, T)}.
Let Y be a random variable defined over S that assigns to
each outcome the number of heads
Example 2
Let S be the sample space of an experiment consisting of
tossing two fair coins. Then, S = {(H, H}, (H, T), (T, H), (T, T)}.
Let Y be a random variable defined over S that assigns to
each outcome the number of heads. Then, Y is a random
variable that takes on values 0, 1, 2:
P(Y =0) = 1/4
P(Y =1) = 2/4
P(Y=2) = 1/4.
P(Y =0) + P(Y =1) + P(Y=2) = 1.
Example 3
A die is repeatedly tossed until a six appears. Let X denote
the number of tosses required, assuming successive tosses are
independent.
Example 3
A die is repeatedly tossed until a six appears. Let X denote
the number of tosses required, assuming successive tosses are
independent. The random variables X takes on values 1, 2, ...,
with respective probabilities:
Example 3
A die is repeatedly tossed until a six appears. Let X denote
the number of tosses required, assuming successive tosses are
independent. The random variables X takes on values 1, 2, ...,
with respective probabilities:
P(X=1) = 1/6
P(X=2) = (5/6)(1/6)
P(X=3) = (5/6)2(1/6)
P(X=4) = (5/6)3(1/6)
.
.
.
P(X=n) = (5/6)n-1(1/6)
P



{ X  n}   i 1 P ( X  n)
n 1

1 
1 n 1
=  i 1 (1  )
6
6
1/6
=
1-(1-1/6)
=1
Distribution functions
The distribution function F (also called cumulative
distribution function (cdf)) of a random variable is
defined by F(x) = P(X ≤ x), where x is a real number.
Distribution functions
The distribution function F (also called cumulative
distribution function (cdf)) of a random variable is
defined by F(x) = P(X ≤ x), where x is a real number.
upper case
lower case
Properties of distribution functions
(i) F ( x) is a nondecreasing function of x.
.
Properties of distribution functions
(i) F ( x) is a nondecreasing function of x.
(ii) lim F ( x)  F () =1.
x 
Properties of distribution functions
(i ) F ( x) is a nondecreasing function of x.
(ii ) lim F ( x)  F () =1.
x 
(iii ) lim F ( x)  F ( ) =0.
x 
Properties of distribution functions
(i) F ( x) is a nondecreasing function of x.
(ii) lim F ( x)  F () =1.
x 
(iii) lim F ( x)  F ( ) =0.
x 
(iv) P( X  x)  1  P( X  x)  1  F ( x).
.
Properties of distribution functions
(i ) F ( x) is a nondecreasing function of x.
(ii ) lim F ( x)  F () =1.
x 
(iii ) lim F ( x )  F ( ) =0.
x 
(iv ) P ( X  x)  1  P ( X  x)  1  F ( x).
(iv ) P ( a  X  b)  P ( X  b)  P ( X  b)  F (b)  F (a ).
Note that P(X < x) does not necessarily equal F(x) since F(x)
includes the probability that X equals x.
Discrete random variables
A discrete random variable is a random variable that
takes on either a finite or a countable number of
states.
Probability mass function
The probability mass function (pmf) of a discrete random
variable is defined by p(x) = P(X=x).
If x takes on values x1, x2, ..., then


i 1
p( xi )  1.
Probability mass function
The probability mass function (pmf) of a discrete random
variable is defined by p(x) = P(X=x).
If x takes on values x1, x2, ..., then


i 1
p( xi )  1.
The cumulative distribution function F is given by
F ( x)   all x  x p( xi ).
i
Example
Let X be a random with pmf p(2) = 0.25, p(4) = 0.6, and p(6) =
0.15. Then, the cdf F of X is given by
Example
Let X be a random with pmf p(2) = 0.25, p(4) = 0.6, and p(6) =
0.15. Then, the cdf F of X is given by
0
 0.25

F ( x)  
0.85

1
if
if
if
if
x2
2 x4
4 x6
6x
The Bernoulli random variable
Let X be a random variable that takes on values 1 (success) or
0 (failure), then the pmf of X is given by
p(0) = P(X=0) = 1 - p and
p(1) = P(X=1) = p
where 0 ≤ p ≤ 1 is the probability of “success.”
A random variable that has the above pmf is said to be a
Bernoulli random variable.
The Geometric random variable
A random variable that has the following pmf is said to be a
geometric random variable with parameter p.
p(n) = P(X=n) = (1 – p)n-1p,
for n = 1, 2, ... .
The Geometric random variable
A random variable that has the following pmf is said to be a
geometric random variable with parameter p.
p(n) = P(X=n) = (1 – p)n-1p,
for n = 1, 2, ... .
Example: A series of independent trials, each having a
probability p of being a success, are performed until a success
occurs. Let X be the number of trials required until the first
success.
The Binomial random variable
A random variable that has the following pmf is said to be a
geometric random variable with parameters (n, p)
n i
p(i)    p (1  p) n i
i
n
n!
where    n Ai 
,
(n  i )!i !
i
n is an integer  1 and 0  p  1.
Example: A series of n independent trials, each having a
probability p of being a success and 1 – p of being a failure,
are performed until a success occurs. Let X be the number of
successes in the n trials.
The Poisson random variable
A random variable that has the following pmf is said to be a
Poisson random variable with parameter
l (l  0)
i
p (i )  P{ X  i}  e
l
l
i!
, i  0,1,....
The number of cars sold per day by a dealer is Poisson with
parameter l = 2. What is the probability of selling no cars
today? What is the probability of receiving 100?
Solution:
P(X=0) = e-2  0.135
P(X = 2)= e-2(22 /2!)  0.270
Example: The number of cars sold per day by a dealer is
Poisson with parameter l = 2. What is the probability of
selling no cars today? What is the probability of receiving
100?
Solution:
P(X=0) = e-2  0.135
P(X = 2)= e-2(22 /2!)  0.270
Continuous random variables
A continuous random variable is a random variable
whose set of possible values is uncountable.
In particular, we say that
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