INTERNATIONAL UNIVERSITY (IU)
Engineering Probability & Statistic ISE Department
Lecturer: Phan Nguyễn Kỳ Phúc
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DISCRETE RANDOM VARIABLES 1. Content
 Discrete Random Variables (RV) o Expectation o Variances  Some useful discrete RV
2. Discrete Variable  Properties 1 P(x) 0  , x   P(x) 1  all x
 The cumulative distribution function, F(x), of a discrete random variable X is: F( ) x  ( P X  ) x   ( P ) i all i x 
 The expected value (or
expectation) of a random variable is the value that is expected to occur, on average.
E(x)  xP(x) all x
 The variance of a random variable is the expected squared deviation from the mean: 2 2 2  V  (X ) E
 [(X - ) ] (x -) P(x) all x 2     2 2 2 2   ( E X ) -[ (
E X)]   x ( P ) x  -   ( xP ) x   all x   all x 
 The standard deviation of a random variable is the square root of its variance:  s
 td ( X )  Var( X )
 The expected value of a function of a discrete random variable X is:
E[h(X )] h
 (x)P(x) all x
 The expected value of a linear function of a random variable is:
E(aX  b) a
 E( X ) b
 The variance of a linear function of a random variable is: 2 2 2
V (a X b) a
 V (X ) a  
 The mean or expected value of the is the sum of their means or
sum of random variables expected values: 
E(X Y ) E(X )  E(Y )      ( X Y) X Y 03 Discrete Random Variable 1
INTERNATIONAL UNIVERSITY (IU)
Engineering Probability & Statistic ISE Department
Lecturer: Phan Nguyễn Kỳ Phúc
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 The VARIANCE of the SUM of independent random variables is the SUM of their VARIANCES: 2 2 2  V
 ( X Y ) V
 ( X ) V (Y )     ( X Y  ) X Y Example:
It is forecast that the total market demand for an item will follow 1 of 6 scenarios with
corresponding probabilities as follows: Scenario Demand Probabilitie s 1 1000 0.1 2 2000 0.15 3 3000 0.25 4 4000 0.25 5 5000 0.15 6 6000 0.1
1. Find the expected value and the variance of the demand of this market
2. To satisfy the customer demand, the company has to invest a production line cost about 1000.
Assume that they can realize the scenario before manufacturing, i.e. the production quantity is
equal to total customers’ demands, and each demand will create a profit of 2. What is the
expected profit of the company?
3. Now assume that, the company must manufacture before realizing market demands. In this
case if they manufacture more than market demands, each unsold product will suffer a loss of
0.5. So among 1000, 2000, 3000, 4000, 5000 and 6000, what is the optimal quantity for manufacturing.
1. Using the formula for calculating the expected values E(x)  xP  (x) all x Scenario Demand Probabilitie D×Prob D2×Prob s 1 1000 0.1 100 100000 2 2000 0.15 300 600000 3 3000 0.25 750 2250000 4 4000 0.25 1000 4000000 5 5000 0.15 750 3750000 6 6000 0.1 600 3600000
The expected value of the demand is the sum of the column (D×Prob) 03 Discrete Random Variable 2
INTERNATIONAL UNIVERSITY (IU)
Engineering Probability & Statistic ISE Department
Lecturer: Phan Nguyễn Kỳ Phúc
--------------------o0o------------------  E(Demand) = 3500
To calculate the variance of the demand, the following formula is applied 2     2 2 2 2   ( E X ) -[ (
E X)]   x ( P ) x -  ( xP ) x 2050000       all x   all x 
2. The expected profit of the company
E h(x )  [ E 2X  1000] 2
 E( X )  1000 6  000 3.
Let Y be the manufacturing quantity of the company: Y={1000, 2000, … 6000};
Let X be the real demand of the market: X={1000, 2000, … 6000};
When X>Y, i.e. demand > manufacturing quantity, there are no unsold items. The number of sold
items is Y and will create a profit of 2Y. The loss due to unsold items is 0
When Xwill create a profit of 2X. The unsold items will incur a cost which is equal to 0.5(Y-X).
So given a pair (X,Y), the profit will be
Profit = 2×min(X,Y)-0.5×max(0,Y-X)
The expected profit of the company No Deman Prob Y=1000 Y=2000 Y=3000 Y=4000 Y=5000 Y=6000 d 1 X=100 0.1 0 200 150 100 50 0 -50 2 X=200 0.15 0 300 600 525 450 375 300 3 X=300 0.25 0 500 1000 1500 1375 1250 1125 4 X=400 0.25 0 500 1000 1500 2000 2000 2000 5 X=500 0.15 0 300 600 900 1200 1500 1425 6 X=600 0.1 0 200 400 600 800 1000 1200 2000 3750 5125 5875 6125 6000 03 Discrete Random Variable 3
INTERNATIONAL UNIVERSITY (IU)
Engineering Probability & Statistic ISE Department
Lecturer: Phan Nguyễn Kỳ Phúc
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Based on this table we obtain the optimal position is Y=5000 03 Discrete Random Variable 4
INTERNATIONAL UNIVERSITY (IU)
Engineering Probability & Statistic ISE Department
Lecturer: Phan Nguyễn Kỳ Phúc
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Some useful Discrete RV Binomial Distribution
Hypergeometric Distribution Poisson Distribution Definition
Measure the probability of X successes Measure the probability of X
Measure the probability of a number of
in n trials where the probability of
successes in n selections, without
occurrences over a given period of success in each trial is p
replacement, from a population of
time or within a given area or volume. N elements, S of which are
That is, the Poisson random variable
successes and (N-S) of which are
counts occurrences over a continuous failures
interval of time or space. It can also
be used to calculate approximate
binomial probabilities when the
probability of success is small (p≤0.05)
and the number of trials is large (n≥20). Formula  n
 S   N  S  x     n x n x ! e ( ) x (n x ) ( P ) x  p q    p q     ( P ) x  for x = 1,2,3,... x ! x ( n    ) x !   X  
  n  x  x ! ( P ) x   q  N  1   p    n    E(X) E(X )   n  p S E X     , np where p  ( ) N Var(X) 2 Var( X )   n  pq 2 N  n Var( X )   Var (X )        npq  N  1 03 Discrete Random Variable 5
INTERNATIONAL UNIVERSITY (IU)
Engineering Probability & Statistic ISE Department
Lecturer: Phan Nguyễn Kỳ Phúc
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Multinomial Distribution Geometric Distribution Definition The multinomial probability
Geometric random variable
distribution is useful for determining
counts the number of trials until
the probabilities of outcomes that can the first success.
be classified into more than two categories. Formula ! n 1 x x x ( ) x P x pq   1 2 2 (
P x , x ,.., x )  p p ... p 1 2 k 1 2
x ! x !...x ! k 1 2 k
n x  x ...  x 1 2 k 1 p  p .  ..  p 1 2 k E(X)
E(X )  np 1   p Var(X) 2 Var(X )   n  pq 2 q Var (X )    2 p 03 Discrete Random Variable 6
INTERNATIONAL UNIVERSITY (IU)
Engineering Probability & Statistic ISE Department
Lecturer: Phan Nguyễn Kỳ Phúc
--------------------o0o------------------ Assignments
Question 1: There are 4 Red balls and 6 Blue balls in a box. Find the probability that we obtain 2
Red balls and 2 Blue balls after 4 drawings when we use 2 different methods as follows:
a) After each draw, we don’t return any ball into the box
b) After each draw, a ball with the same color is returned into the box
c) Now we draw until there is no ball in the box. What is the probability that the last ball is the Red ball.
Question 2: The following gambling game, known as the wheel of fortune (or chuck-a-luck), is
quite popular at many carnivals and gambling casinos: A player bets on one of the numbers 1
through 6. Three dice are then rolled, and if the number bet by the player appears i times, i = 1, 2,
3, then the player wins i units; if the number bet by the player does not appear on any of the dice,
then the player loses 1 unit. Is this game fair to the player? (Actually, the game is played by
spinning a wheel that comes to rest on a slot labeled by three of the numbers 1 through 6, but this
variant is mathematically equivalent to the dice version.)
Question 3: A communication system consists of n components, each of which will,
independently, function with probability p. The total system will be able to operate effectively if
at least one-half of its components function. For what values of p is a 5-component system more
likely to operate effectively than a 3-component system?
Question 4:To determine whether they have a certain disease, 100 people are to have their blood
tested. However, rather than testing each individual separately, it has been decided first to place
the people into groups of 10. The blood samples of the 10 people in each group will be pooled
and analyzed together. If the test is negative, one test will suffice for the 10 people, whereas if
the test is positive, each of the 10 people will also be individually tested and, in all, 11 tests will
be made on this group. Assume that the probability that a person has the disease is .1 for all
people, independently of each other, and compute the expected number of tests necessary for
each group. (Note that we are assuming that the pooled test will be positive if at least one person in the pool has the disease.)
Question 5: A newsboy purchases papers at 10 cents and sells them at 15 cents. However, he is
not allowed to return unsold papers. If his daily demand is a binomial random variable with n =
10, p = 1/3 approximately how many papers should he purchase so as to maximize his expected profit?
Question 6: It is known that screws produced by a certain company will be defective with
probability .01, independently of each other. The company sells the screws in packages of 10 and
offers a money-back guarantee that at most 1 of the 10 screws is defective. What proportion of
packages sold must the company replace? 03 Discrete Random Variable 7
INTERNATIONAL UNIVERSITY (IU)
Engineering Probability & Statistic ISE Department
Lecturer: Phan Nguyễn Kỳ Phúc
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Question 7: An airline knows that 5% of the people making reservations on a certain flight will
not show up. Consequently, their policy is to sell 52 tickets for a flight that can hold only 50
passengers. What is the probability that there will be a seat available for every passenger who shows up.
Question 8: ,
In a game show the player will be rewarded 100 dollars for each successful answer.
And he has to stop when he has 1st wrong answer. Assumed that all question has 4 possible
answers, and the player only uses blind guess, what is the expected money quantity that a player can obtain?
Question 9: If the condition in a day is favorable, fungi will double number of themselves in that
day. Otherwise, the quantity of fungi won’t change. Assumed that at the beginning of day 1 we
have 3 fungi, what is the expected number of fungi after 4 days? Given that the probability a day is a favorable day is 0.6. Question 10
: Two athletic teams play a series of games; the first team to win 4 games is declared
the overall winner. Suppose that one of the teams is stronger than the other and wins each game
with probability .6, independently of the outcomes of the other games. Find the probability, for i
= 4, 5, 6, 7, that the stronger team wins the series in exactly i games. Compare the probability
that the stronger team wins with the probability that it would win a 2-out-of-3 series.
Question 11: A contractor purchases a shipment of 100 transistors. It is his policy to test 10 of
these transistors and to keep the shipment only if at least 9 of the 10 are in working condition. If
the shipment contains 20 defective transistors, what is the probability it will be kept?
Question 12: When three friends go for coffee, they decide who will pay the check by each
flipping a coin and then letting the “odd person” pay. If all three flips produce the same result (so
that there is no odd person), then they make a second round of flips, and they continue to do so
until there is an odd person.What is the probability that
1. Exactly 3 rounds of flips are made?
2. More than 4 rounds are needed?
Question 13: Each of the members of a 7-judge panel independently makes a correct decision
with probability. 7. If the panel’s decision is made by majority rule, what is the probability that
the panel makes the correct decision? Given that 4 of the judges agreed, what is the probability
that the panel made the correct decision?
Question 14: The following gambling game, known as the wheel of fortune (or chuck-a-luck), is
quite popular at many carnivals and gambling casinos: A player bets on one of the numbers 1
through 6. Three dice are then rolled, and if the number bet by the player appears i times, i = 1, 2,
3, then the player wins i units; if the number bet by the player does not appear on any of the dice, 03 Discrete Random Variable 8
INTERNATIONAL UNIVERSITY (IU)
Engineering Probability & Statistic ISE Department
Lecturer: Phan Nguyễn Kỳ Phúc
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then the player loses 1 unit. Is this game fair to the player? (Actually, the game is played by
spinning a wheel that comes to rest on a slot labeled by three of the numbers 1 through 6, but this
variant is mathematically equivalent to the dice version.
Question 15: A game show consists of 2 rounds. Obviously, to advance into the 2 round, the nd
player must win the 1st round first. The probabilities that the player wins the 1st and the 2 nd
round are given as 0.60 and 0.40, respectively. During the 1st round, the player will be rewarded
$10 for winning, and $5 for losing (considered as reward for participating). The player will be
rewarded $20 for winning, and $8 for losing. This reward is independent with the reward of the 1st round.
1. Find the probability that the player win both rounds.
2. Given that the player has lost, find the probability that the player lost at 2 round. nd
3. Find the expected reward money that the player receives in this gameshow.
Question 16: In a court action, jury is a group of 12 people – which call juror, that votes to see if
a defendant is guilty or not. Suppose that a defendant is considered guilty if at least 10 out of 12
jury members vote guilty. Each jury member votes independently of each other. A juror has
probability that vote a guilty defendant innocent is 0.25, and the probability that a juror votes an
innocent defendant is 0.05. Given that on today, 75% of the defendants are actually guilty.
Calculate the probability that the jury provide a correct decision?
Question 17: The number of typos in a page follows Poisson distribution with µ=8.
1. What is the probability that no typos in a page?
2. What is the probability that at least 4 typos in a page.
Question 18: People enter a casino with a rate of 1 people in 2 minutes.
1. What is the probability that no one enters between 12:00 and 12:05?
2. What is the probability that at least 4 people enter the casino during that time. 03 Discrete Random Variable 9