Chapter Two Discrete Random Variable

Example 2.1-1 A rat is selected at random from a cage and its sex is determined.

Definition 2.1-1

Given a random experiment with an outcome space S, a function X that assigns one and only one
real number X(s) = x to each element s in S is called a random variable. The space of X is the set
of real numbers {x : X(s) = x, s ∈ S}, where s ∈ S means that the element s belongs to the set
S.

Example 2.1-2: Let the random experiment be the cast of a die.

The student will no doubt recognize two major difficulties here:

1. In many practical situations, the probabilities assigned to the events are unknown.
2. Since there are many ways of defining a function X on S, which function do we want to use?
Example2.1-3: Roll a fair four-sided die twice, and let X be the maximum of the two outcomes.

Two types of graphs can be used to give a better visual appreciation of

the pmf:
A line graph of the pmf f (x)

A probability histogram
Consider a collection of N = N1+N2 similar objects, N1 of them belonging to one of two dichotomous
classes (red chips, say) and N2 of them belonging to the second class (blue chips, say). A collection
of n objects is selected from these N objects at random and without replacement. Find the
probability that exactly x (where the nonnegative integer x satisfies x ≤ n, x ≤ N1, and n − x ≤ N2)

We say that the random variable X has a hypergeometric distribution.

Example 2.1-5: In a small pond there are 50 fish, 10 of which have been tagged. If a fisherman’s
catch consists of 7 fish selected at random and without replacement, and X denotes the number of
tagged fish, what is the probability that exactly 2 tagged fish are caught?

Example 2.1-6: A lot (collection) consisting of 100 fuses is inspected by the following procedure:
Five fuses are chosen at random and tested; if all five blow at the correct amperage, the lot is
accepted. Suppose that the lot contains 20 defective fuses. If X is a random variable equal to the
number of defective fuses in the sample of 5, what is the probability of accepting the lot?

Example 2.1-7: A fair four-sided die with outcomes 1, 2, 3, and 4 is rolled twice. Let X equal the
sum of the two outcomes. Then the possible values of X are 2, 3, 4, 5, 6, 7, and 8. The following
argument suggests that the pmf of X is given by f (x) = (4 − |x − 5|)/16, for x = 2, 3, 4, 5, 6, 7, 8
2.1-2: A bowl contains 6 white, 3 red 1 blue. Draw one ball randomly from the box. Let the RV

. Obtain the pdf of X

2.1-3: Find such that

HW ( a ) f(x) = x/c, x = 1, 2, 3, 4

H.W( d ) f(x) = c(x + 1)2, x = 0,1,2,3

HW ( e ) f(x) = x/c, x = 1, 2, 3, . . . , n

b)

c)

f)
2.1-8:A fair four-sided die has two faces numbered 0 and two faces numbered 2. Another fair four-
sided die has its faces numbered 0, 1, 4, and 5. The two dice are rolled. Let X and Y be the
respective outcomes of the roll. Let W = X + Y. Determine the pmf of W
0 2
0 0 2
1 1 3
4 4 6
5 5 7
Probabilities

2.1-10:Suppose there are 3 defective items in a lot (collection) of 50 items. A sample of size 10 is
taken at random and without replacement. Let X denote the number of defective items in the
sample. Find the probability that the sample contains
(a) Exactly one defective item

(b) At most one defective item

(HW)2 .1 -11. In a lot (collection) of 100 light bulbs, there are 5defective bulbs. An inspector
inspects 10 bulbs selected atrandom. Find the probability of finding at least one defective bulb.
Hint: First compute the probability of findingno defectives in the sample.
(HW)2 .1-12 . Let X be the number of accidents per week in afactory. Let the pmf of X be

Find the conditional probability of X ≥ 4, given thatX ≥ 1.

(HW)2 .1-13 . A professor gave her students six essay questionsfrom which she will select three for
a test. A student hastime to study for only three of these questions. What isthe probability that,
of the questions studied,
(a) at least one is selected for the test?

(b) all three are selected?

(c) exactly two are selected?

Class(2.1): 2, 3(b,c,f),8, 10, 17

HW(2.1): 3(a,d,e), 11, 12, 13

2.2 Mathematical Expectation

(2.2)HW: 2, 12
Class: 3

Example 2.2-1: An enterprising young man who needs a little extra money devises a game of
chance in which some of his friends might wish to participate. The game that he proposes is to let
the participant cast a fair die and then receive a payment according to the following schedule: If the
event A = {1, 2, 3} occurs, he receives one dollar; if B = {4, 5} occurs, he receives two dollars; and
if C = {6} occurs, he receives three dollars. If X is a random variable that represents the payoff,
what is the pmf of X? what is the average payment?

Note that this mathematical expectation can be written . and is often denoted by the
Greek letter , which is called the mean of X or of its distribution.

Example 2.2-2: Let the random variable X have the pmf let . Find
Example 2.2-3: Let X have the pmf find

a)

b)

c)

(HW)2.2-2. Let the random variable X have the pmf f(x) = (|x| + 1)2 /9, x = −1, 0, 1.
Compute E(X)

E(X2 )

and E(3X2 − 2X + 4)
2.2-3. Let the random variable X be the number of days that a certain patient needs to be in the
hospital. Suppose X has the pmff(x)=(5–x)/10, x=1,2,3,4. If the patient is to receive $200 from an
insurance company for each of the first two days in the hospital and $100 for each day after the first
two days, what is the expected payment for the hospitalization?
X
1
2
3
4

(HW)2.2-12. Suppose that a school has 20 classes: 16 with 25 students in each, three with 100
students in each, and one with 300 students, for a total of 1000 students.
(a) What is the average class size?

(b) Select a student randomly out of the 1000 students. Let the random variable X equal the size
of the class to which this student belongs, and define the pmf of X.

(c) Find E(X), the expected value of X. Does this answer surprise you?
2.3 Special mathematical expectations
(2.3)HW: 1, 4, 19(a,b)
Class: 3, 13, 11
 The mean of the random variable X (or of its distribution) is
 The variance of the random variable X (or of its distribution) is

 The positive square root of the variance is called the standard deviation of X and is denoted
by the Greek letter (sigma).
 The second moment about the origin is
Example 2.3-1: Let X equal the number of spots on the side facing upward after a fair six-sided die
is rolled. A reasonable probability model is given by the pmf
Find :
The mean

The second moment about the origin

The variance

The standard deviation

Example 2.3-2: Let X have the pmf f (x) = 1/3, x = −1, 0, 1. Find the mean and the variance of X.
Example 2.3-3: Let X have a uniform distribution on the first m positive integers.

The mean of X is
The variance of X is

For example, we find that if X equals the outcome when rolling a fair six-sided
die, the pmf of X is
the respective mean and variance of X are

 The rth moment of the distribution about the origin is where r be a positive
integer.
 the rth moment of the distribution about b is

M(0) = 1. Moreover, if the space of S is {b1, b2, b3, . . .}, then the moment-generating function is
given by the expansion

Thus, the coefficient of is the probability

Example 2.3-5: If X has the mgf

Find S

P(X = 1)
P(X = 2)
P(X = 3)

The pmf
Example 2.3-5: If X has the mgf

Note: , thus,

for each positive r

Setting t = 0, we see that

In particular, if the moment-generating function exists, then

and

(HW) 2.3-1. Find the mean and variance for the following discrete distributions:
(a) f(x) = 1/5 , x = 5, 10, 15, 20, 25.

(b) f(x) = 1, x = 5.

(c) f(x) = (4 – x)/6 , x = 1, 2, 3.

2.3-3. Given E(X+4)= 10 and E[(X + 4)2]=116, determine

(a) Var(X+4)

(b) μ=E(X)

(c) σ 2 = Var(X)

(HW)2.3-4. Let μ and σ 2 denote the mean and variance of the random variable X. Determine
E[(X − μ)/σ] and E{[(X − μ)/σ]2}.

2.3-11. If the moment-generating function of X is , find the mean, variance,

and pmf of X.
 (HW)2.3-19. Given a random permutation of the integers in the set {1, 2, 3, 4, 5}, let X equal the
number of integers that are in their natural position. The moment-generating function of X is

(a) Find the mean and variance of X.

(b) Find the probability that at least one integer is in its natural position.
2.4 Binomial Distribution
(2.4)HW:6, 8, 11, 19 Class: 1,2,3 ,4, 20
The probability models for random experiments that will be described in this section occur frequently
in applications. A Bernoulli experiment is a random experiment, the outcome of which can be
classified in one of two mutually exclusive and exhaustive ways—say, success or failure (e.g., female
or male, life or death, non defective or defective). A sequence of Bernoulli trials occurs when a
Bernoulli experiment is performed several independent times and the probability of success—say,
p—remains the same from trial to trial. That is, in such a sequence we let p denote the probability of
success on each trial. In addition, we shall frequently let q = 1 − p denote the probability of failure;
that is, we shall use q and 1 − p interchangeably.

Bernoulli experiment is any experiment that has exactly two outcomes, say success (s) and fail
(f).

the random variable X is said to have Bernoulli distribution with probability of success P if its p.d.f is
given by

Note (probability of fail).

And (probability of success).

If X has a Bernoulli with probability of success P, , then

1. The mean of X is
2. The variance of X is
3. The standard deviation is
4. The m.g.f is .

Example: A box contains 6 red balls and four white balls. We draw one ball. Let x be a white ball.
Find:
1.

2. The mean of X

3. The variance
 4. The m.g.f

5. The c.d.f

Example 2.4-3: Out of millions of instant lottery tickets, suppose that 20% are winners. If five such
tickets are purchased, then (0, 0, 0, 1, 0) is a possible observed sequence in which the fourth ticket
is a winner and the other four are losers. Assuming independence among winning and losing tickets,
what is the probability of this outcome?

Example 2.4-4: If five beet seeds are planted in a row, a possible observed sequence would be
(1, 0, 1, 0, 1) in which the first, third, and fifth seeds germinated and the other two
did not. If the probability of germination is p = 0.8, what is the probability of this outcome?

Example 2.4-1: Suppose that the probability of germination of a beet seed is 0.8 and the
germination of a seed is called a success. If we plant 10 seeds and can assume that the germination
of one seed is independent of the germination of another seed
In a sequence of Bernoulli trials, we are often interested in the total number of successes but not the
actual order of their occurrences. If we let the random variable X equal the number of observed
successes in n Bernoulli trials, then the possible values of X are 0, 1, 2, . . . , n. If x successes occur,
where x = 0, 1, 2, . . . , n, then n – x failures occur. The number of ways of selecting x positions for
the x successes in the n trials is . Since the trials are independent and since the
probabilities of success and failure on each trial are, respectively, p and q = 1 − p, the probability of
each of these ways is . Thus

These probabilities are called binomial probabilities, and the random variable X is said to have a
binomial distribution, , . Summarizing, a binomial experiment satisfies the following
properties:
1. A Bernoulli (success–failure) experiment is performed n times, where n is a (non-random)
constant.
2. The trials are independent.
3. The probability of success on each trial is a constant p (i.e. the P remain the same from trial to
trial); the probability of failure is q = 1 − p.
4. The random variable X equals the number of successes in the n trials.

If X has a Binomial with probability of success P, , then

1. The mean of X is
2. The variance of X is
3. The standard deviation is
4. The m.g.f is .

Example 2.4-5: In the instant lottery with 20% winning tickets, ifX is equal to the number of
winning
tickets among n = 8 that are purchased, then the probability of purchasing two winning tickets is

Example 2.4-7: In Example 2.4-1, the number X of seeds that germinate in n = 10 independent
trials is
Example: Draw 5 balls at random with replacement from a box contains 10 red and 20 white balls.
Let X is the number of red balls. Find:
1. The probability that you get 3 red balls.

2. The probability that you get at least 3 red balls.

3. The probability that you get at least 3 red balls.

For the binomial distribution b(n, p) distribution, the comulative distribution function is defined by

For the last example:

P(X ≤ 8) =

P(X ≤ 6) =

Example: A box contains 10 red and 15 white balls. If we draw 8 balls. Let X the number of red
balls. Find P(X =2) if
1. Drawn with replacement.

2. Drawn without replacement.

2.4-1. An urn contains 7 red and 11 white balls. Draw one ball at random from the urn. Let X = 1 if
a red ball is drawn, and let X = 0 if a white ball is drawn. Give the pmf, mean, and variance of X.

2.4-2. Suppose that in Exercise 2.4-1, X = 1 if a red ball is drawn and X = −1 if a white ball is
drawn. Give the pmf, mean, and variance of X.

2.4-3. On a six-question multiple-choice test there are five possible answers for each question, of
which one is correct (C) and four are incorrect (I). If a student guesses randomly and
independently, find the probability of

(a) Being correct only on questions 1 and 4 (i.e., scoring C, I, I, C, I, I).

(b) Being correct on two questions.

2.4-4. It is claimed that 15% of the ducks in a particular region have patent schistosome infection.
Suppose that seven ducks are selected at random. Let X equal the number of ducks that are
infected.
(a) Assuming independence, how is X distributed?

(b) Find (i) P(X ≥ 2), (ii) P(X = 1), and (iii) P(X ≤ 3)

(HW)2.4-6. It is believed that approximately 75% of Americanyouth now have insurance due to the
health care law.Suppose this is true, and let X equal the number ofAmerican youth in a random
sample of n = 15 withprivate health insurance.
(a) How is X distributed?

(b) Find the probability that X is at least 10.

(c) Find the probability that X is at most 10.

(d) Find the probability that X is equal to 10.

(e) Give the mean, variance, and standard deviation of X

(HW)2.4-8. A boiler has four relief valves. The probability that each opens properly is 0.99.
(a) Find the probability that at least one opens properly.

(b) Find the probability that all four open properly.

(HW)2.4-11. A random variable X has a binomial distribution with mean 6 and variance 3.6. Find
P(X = 4)
(HW)

THE GEO MATRIC DIS TRIBUTION:

In a random experiment, if the probability of success is P and probability of failure is q. If we
repeat the Bernoulli trial till we get the first success, then we define the r.v X as the number of
trials before the first success. Thus the p.mf is

And we called it geometric distribution and denoted by

1. The mean of X is
2. The variance of X is

3. The standard deviation is

4. The m.g.f is
5.
6.
7. note this is the c.d.f
8.
Example: let . find:
1. The p.m.f

2. The mean

3. The variance

4. The m.g.f

5.

6.

7.

8.
 9. The probability that at least six trials are needed to get the first success

10. The probability that at most five trials are needed to get the first success

Example: let X has a geometric distribution w ith mean 4. Find the variance.

2.3-13. For each question on a multiple-choice test, there are five possible answers, of which
exactly one is correct. If a student selects answers at random, give the probability that the first
question answered correctly is question 4.
DIS TRIBUTION MEAN VARIANC E
2.5 Negative Binomial distribution
2.5(HW):2, 4 Class:1, 3, 8, 9

If we repeat a Bernoulli trials till we get r success. Let the r.v X denoted the number of trials needed
to observe the rth success. The p.mf of X is

We say that x has a negative binomial distribution note if r=1 X has a geometric
distribution.
1. The mean of X is
2. The variance of X is

3. The standard deviation is

4. The m.g.f is

Example 2.5-2: Suppose that during practice a basketball player can make a free throw 80% of the
time. Furthermore, assume that a sequence of free-throw shooting can be thought of as independent
Bernoulli trials. Let X equal the minimum number of free throws that this player must attempt to
make a total of 10 shots. The pmf of X is

2.5-1. An excellent free-throw shooter attempts several free throws until she misses.
(a) If p = 0.9 is her probability of making a free throw, what is the probability of having the first
miss on the 13th attempt or later?

(b) If she continues shooting until she misses three, what is the probability that the third miss
occurs on the 30th attempt?
(HW)2.5-2. Show that 63/512 is the probability that the fifth head is observed on the tenth
independent flip of a fair coin.

2.5-3. Suppose that a basketball player different from theones in Example 2.5 -2 and in Exercise
2.5-1 can make afree throw 60% of the time. Let X equal the minimumnumber of free throws that
this player must attempt to make a total of 10 shots.
(a) Give the mean, variance, and standard deviation of X.

(b) Find P(X = 16).

(HW)2.5-4. Suppose an airport metal detector catches a personwith metal 99% of the time. That
is, it misses detectinga person with metal 1% of the time. Assume independence of people carrying
metal. What is the probabilitythat the first metal-carrying person missed (not detected)is among the
first 50 metal-carrying persons scanned?
2.5-8. The probability that a company’s work force hasno accidents in a given month is 0.7. The
numbers of accidents from month to month are independent. What isthe probability that the third
month in a year is the firstmonth that at least one accident occurs?

2.5-9. One of four different prizes was randomly put intoeach box of a cereal. If a family decided to
buy this cerealuntil it obtained at least one of each of the four different prizes, what is the expected
number of boxes of cerealthat must be purchased?