Download Discrete Random Variables and Probability Distributions

Systems Engineering Program
Department of Engineering Management, Information and Systems
EMIS 7370/5370 STAT 5340 :
PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS
Discrete Probability Distributions
Discrete Random Variables &
Probability Distributions
Dr. Jerrell T. Stracener, SAE Fellow
Leadership in Engineering
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Random Variable
Definition - A random variable is a mathematical
function that associates a number with every
possible outcome in the sample space S.
Notation - Capital letters, usually X or Y, are
used to denote random variables. Corresponding
lower case letters, x or y, are used to denote
particular values of the random variables X or Y.
Definition - A discrete random variable X is a
random variable that can take on or assume a
finite number of possible values, say x1, x2, …, xk
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Probability Mass Function
Associated with a discrete random variable X
having possible values x1, x2, …, xn is a function
called the probability mass function. The probability
mass function of X associates with each possible
value of X the probability of its occurrence. This
set of ordered pairs, each of the form,
(value of x, probability of that value occurring)
or
( x, p(x) )
is the probability mass function of X.
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Probability Mass Function
The function p (x )is the probability mass function
of the discrete random variable X if, for each
possible outcome ,
x
1.
p ( x)  0
2.
 p( x)  1
X
3.
P( X  x)  p ( x)
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Probability Distribution Function
The (cumulative) probability distribution function,
F (x), of a discrete random variable Xwith
probability mass function p (x )is given by
F ( x)  P( X  x)
  p(t )
t X
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p(x)
Probability
Mass
Function
x
0
1
2
3
4
F(x) 1
Probability
Distribution
Function
0.5
0
x
0
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3
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Example - Probability Mass Function
and Probability Distribution Function
If an experiment is “Toss a coin 3 times in sequence”
and the random variable X is defined to be the number
of heads that result, determine and plot the probability
mass function and probability distribution function for X
if
(a) The coin is fair
(b) The coin is biased with P(H)=0.75
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Example Solution - Probability Mass Function and
Probability Distribution Function
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Mean or Expected Value of a Discrete
Random Variable X
• Mean or Expected Value of X
μ  EX    xp(x)
all x
•Note:
The interpretation of μ:
The average of X in the long term.
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Example-Calculation of Mean
If an experiment is “Toss a coin 3 times in sequence”
and the random variable X is defined to be the number
of heads that result, what is the mean or expected value
of X if
(a) The coin is fair
(b) The coin is biased with P(H)=0.75
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Example Solution - Calculation of Mean
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Variance & Standard Deviation of a Discrete
Random Variable X
• Variance
– Definition
Var X   σ   (x  μ) p(x)
2
– Rule
2
all x
 
Var X   E X  μ
2
2
  x px   μ
2
2
x
• Standard Deviation
σ  Var(X)
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Example-Calculation of Standard Deviation
If an experiment is “Toss a coin 3 times in sequence”
and the random variable X is defined to be the number
of heads that result, what is the standard deviation of X if
(a) The coin is fair
(b) The coin is biased with P(H)=0.75
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Example – Family Planning
In planning a family of 4 children, find the probability
distribution of:
a.
b.
X = the number of boys
Y = the number of changes in sex sequence
Find (i) the probability mass and distribution functions (and
plot), (ii) the mean, (iii) the variance, and (iv) the standard
deviation.
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Discrete Uniform Distribution
Definition - If the random variable X assumes the
values x1, x2, ... xk with equal probabilities, then X
has a discrete uniform distribution with probability
mass function
1
p( x; k ) 
k
for x  x1 , x 2 , ... x k
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Discrete Uniform Distribution
If X has the discrete uniform distribution U(k), then
the mean and variance are
k
  Ex  
 xi
i 1
k
k
and
 
2
 x
i 1
 
2
i
k
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Rules
If a and b are constants and if  = E(X) is the mean
and 2 = Var(X) is the variance of the random
variable X, respectively, then
EaX  b  aμ  b
and
Var aX  b  a Var X 
2
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Rules
If Y = g(X) is a function of a discrete random variable
X, then
μ Y  Eg x    gx px 
all X
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Chebyshev’s Theorem
The probability that any random variable X will
assume a value within k standard deviations of the
mean is at least
1  1 2 , i.e.,
k
P  k  X    k   1  1
k
2
Remark: Chebyshev’s Theorem gives a conservative
estimate of the probability that a random variable
assumes a value within k standard deviations of its
mean for any real number k.
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