Download 1 Chapter 3 Random Variables and Probability Distributions

Probability and Statistical Methods
Math 322
Chapter 3 Random Variables and Probability Distributions
In statistics we deal with random variables- variables whose observed value is determined by
chance. Random variables usually fall into one of two categories: discrete or continuous
Random Variable. A random variable (r.v.) is a function that associates a real number with each
element in the sample space. Random variables will be denoted by uppercase letters and their
observed numerical values by lowercase letters.
Discrete Random Variable. A random variable is discrete if it can assume at most a finite or a
countably infinite number of possible values.
Example 1. Two balls are drawn in succession without replacement from an urn containing 4 red
and 3 black balls. The possible outcomes and values y of the random variable Y where Y is the
number of red balls are:
Sample Space y
RR
2
RB
1
BR
1
BB
0
Continuous Random Variable. A random variable is continuous if it can assume any value in
some interval or intervals of real numbers and the probability that it assumes any specific value
is 0.
Discrete Probability Distributions
Definition. The set of ordered pairs
( x, f ( x ) )
is a probability function, probability mass
function or probability distribution of the discrete random variable X if,
(1) f ( x ) ≥ 0
(2)
∑ f ( x) = 1
x
(3) P ( X = x ) = f ( x ) .
Example 1. A committee of size 5 is to be selected at random from 3 chemists and 5
mathematicians. Find the probability distribution (p.d.) for the number of chemists on the
committee.
Sonuç Zorlu
Lecture Notes
1
Let X be the number of chemists on the committee. Then x : 0,1, 2,3 .
3 5
  
0 5
1
;
P ( X = 0) = f ( 0) =     =
56
8
 
5
 3  5 
  
1 4
15
P ( X = 1) = f (1) =     =
56
8
 
 5
3 5
  
2 3
30
P ( X = 2) = f ( 2) =     =
;
P ( X = 3) = f ( 3) =
56
8
 
5
Therefore the probability distribution of X is
x
f ( x)
0
1
56
1
15
56
2
30
56
 3  5 
  
 3   2  = 10
56
8
 
 5
3
10
56
Exercise. Among 10 applicants for an open position 6 are female and are males. Suppose 3
applicants are randomly selected from the applicant pool for final interviews. Find the
probability distribution for X , which is the number of female applicants among the final three.
Exercise. Let w be a random variable giving the number of heads minus the number of tails in
three tosses of a coin.
(a) List the elements of the sample space
(b) Assign a value w of W to each sample points.
(c) Find the probability distribution of the random variable W assuming that the coin is
biased so that a head is twice as likely to occur as a tail.
Cumulative Distribution. The cumulative distribution F ( x ) of a discrete random variable X
with probability distribution f ( x ) is
F ( x ) = P ( X ≤ x ) = ∑ f ( t ) , for −∞ ≤ x ≤ ∞ .
t≤x
Example 2. Find the cumulative distribution of the number of red balls in example 1. Using
10
F ( x ) , show that f ( 3) =
.
56
F ( 0) = f ( 0) =
1
;
56
Sonuç Zorlu
Lecture Notes
2
F (1) = f ( 0 ) + f (1) =
16
56
46
;
56
F ( 3) = f ( 0 ) + f (1) + f ( 2 ) + f ( 3) = 1 .
F ( 2 ) = f ( 0 ) + f (1) + f ( 2 ) =
Hence,
if x < 0
0,
1

 56
 16
F ( x) = 
 56
 46
 56

1
46 10
= .
Now, f ( 3) = F ( 3) − F ( 2 ) = 1 −
56 56
if 0 ≤ x < 1
if 1 ≤ x < 2
if 2 ≤ x < 3
if x ≥ 3
Continuous Probability Distributions
The function f ( x ) is a probability density
Definition. (Probability Density Function)
function for the continuous random variable X , defined over the set of real numbers » , if
(1) f ( x ) ≥ 0,
for all x ∈ »
∞
(2)
∫ f ( x ) dx = 1
−∞
b
(3) P ( a < X < b ) = ∫ f ( x ) dx .
a
Note that for a continuous random variable X ,
a
P ( X = a ) = ∫ f ( x ) dx = 0 .
a
Cumulative Distribution. The cumulative distribution F ( x ) of a continuous random variable
X with density function f ( x ) is
F ( x) = P ( X ≤ x) =
x
∫ f ( t )dt
for − ∞ < x < ∞ .
−∞
Sonuç Zorlu
Lecture Notes
3
Example 1. Suppose that a random variable X has a probability density function given by
kx (1 − x ) 0 ≤ x ≤ 1
f ( x) = 
elsewhere
0
(a) Find the value of k that makes this a probability function.
1
∫ kx (1 − x ) dx = 1 ⇒ k = 6
0
(b) Find P ( 0.4 ≤ X ≤ 1)
 ( 0.4 )2 ( 0.4 )3 
∫ 6 x (1 − x ) dx = 1 − 6  2 − 3  = 0.332
0.4


(c) Find F ( x ) = P ( X ≤ x ) and sketch the graph of this function.
1
if x ≤ 0
0,
 2
3
F ( x) = 3 x − 2 x , if 0 < x < 1
1,
if
x ≥1

Example 2. The weekly demand X for kerosene at a certain supply station has a density
function given by
for 0 ≤ x ≤ 1
x

f ( x) = 1/ 2
for 1 < x ≤ 2
0
elsewhere

3
1
(a) Find P  ≤ X ≤ 
2
2
(b) Find F ( x ) = P ( X ≤ x )
Joint Probability Distributions
Joint Probability Distributions of two discrete random variables X and Y :
Definition. The function f ( x, y ) is a joint probability distribution of the discrete random
variables X and Y if
(1) f ( x, y ) ≥ 0 for all ( x, y )
(2)
∑∑ f ( x, y ) = 1
x
y
(3) P ( X = x, Y = y ) = f ( x, y )
For any region A in the xy − plane , P ( X , Y ) ∈ A = ∑∑ f ( x, y )
A
Sonuç Zorlu
Lecture Notes
4
Example 1. Determine the value of c so that the following functions represent the joint
probability distribution of the random variables X and Y .
(a) f ( x, y ) = cxy for x = 1, 2,3, y = 1, 2,3
(b) f ( x, y ) = c x − y for x = −2, 0, 2, y = −2,3
Definition. The function f ( x, y ) is a joint probability density function of the continuous
random variables X and Y if
(1) f ( x, y ) ≥ 0 for all ( x, y )
(2)
∞
∫ ∫
∞
−∞ −∞
f ( x, y )dxdy = 1
(3) P ( X , Y ) ∈ A = ∫∫ f ( x, y )dxdy
A
for any region A in the xy − plane .
Example 2. Let X denote the reaction time, in seconds, to a stimulant and denote Y the
temperature ( 0 F ) at which a certain reaction starts to take place. Suppose that two random
variables X and Y have the joint density
4 xy, 0 < x < 1
f ( x, y ) = 
0, elsewhere
(a) Show that f ( x, y ) is a probability density function
1 1
1

(b) Find P  0 ≤ X ≤ , ≤ Y ≤  .
2 4
2

Marginal Distributions
Definition. The marginal distributions of X alone and Y alone are
g ( x ) = ∑ f ( x, y ) and h ( y ) = ∑ f ( x, y )
y
x
for the discrete case and
g ( x) =
∞
∫ f ( x, y ) dy
and
-∞
h( y) =
∞
∫ f ( x, y ) dx
-∞
for the continuous case.
Example 1. Suppose that X and Y have the following joint probability distribution:
f ( x, y )
y
1
3
5
g ( x)
x
1
2
h( y)
0.10
0.20
0.10
0.40
0.15
0.30
0.15
0.60
0.25
0.50
0.25
1
Sonuç Zorlu
Lecture Notes
5
(a) Find the marginal distribution of X
x
1
g ( x ) 0.40
(b) Find the marginal distribution of Y
y
1
h ( y ) 0.25
2
0.60
3
5
0.50 0.25
Statistical Independence
Definition. Let X and Y be two random variables, discrete and continuous, with joint
probability distribution f ( x, y ) and marginal distributions g ( x) and h( y ) respectively. The
random variables X and Y are said to be statistically independent if and only if
f ( x, y ) = g ( x ) h ( y )
for all ( x, y ) within their range.
Example 1. Let X denote the number of times a certain numerical control machine will
malfunction: 1,2,3 times on any given day. Let Y denote the number of times a technician is
called on an emergency call. Their joint probability distribution is given as
f ( x, y )
y
1
2
3
x
1
2
3
0.05 0.05 0.10
0.05 0.10 0.35
0 0.20 0.10
(a) Find the marginal distribution of X
The Marginal distribution of X is
x
1
2
g ( x)
0.10 0.35
3
0.55
(b) Find the marginal distribution of Y
The marginal distribution of Y is
y
1
h( y )
0.20
3
0.30
2
0.50
(c) Determine whether X and Y are independent or not.
Let X = 1 and Y = 1, then f ( x, y ) = 0.05 . Now check whether f ( x, y ) = g ( x)h( y ) .
?
So, f (1,1) = g (1)h(1) , 0.05 ≠ (0.10)(0.20) , which implies that X and Y are not
independent.
Sonuç Zorlu
Lecture Notes
6
Example 2. Consider the following joint probability density function of the random variables
X and Y :
 3 x − y
, 1 < x < 3,1 < y < 2
f ( x, y ) =  9

elsewhere
0
(a) Find the marginal distributions of X and Y
The marginal distribution of X is
2
3x − y
1 
y 2  2 2 x − 1
, 1< x < 3
g ( x) = ∫
dy = 3 xy −  =
9
9 
2  1
6
1
The marginal distribution of Y is
3
h( y ) = ∫
1
 3 12 − 2 y
3x − y
1  3x 2
, 1< x < 2
dx = 
− yx =
 1
9
9  2
9
(b) Are X and Y independent
If f ( x, y ) = g ( x)h( y ) then X and Y are independent.
Since
3 x − y  2 x − 1
12 − 2 y  , the variables X and Y are not independent.
≠
 9 
 6 
9
(c) Find P ( X > 2) .
3
3
P ( X 2) = ∫ g ( x)dx = ∫
2
2
2 x −1
2
dx = .
6
3
Exercise. A coin is tossed twice. Let Z denote the number of heads on the first toss and W the
total number of heads on the 2 tosses. If the coin is unbalanced and a head has a 40% chance of
occurring, find
(a) the joint probability distribution of W and Z
(b) the marginal distribution of W
(c) the marginal distribution of Z
(d) the probability that at least 1 tail occurs.
Sonuç Zorlu
Lecture Notes
7