CHAPTER 3: Random Variables and Probability Distributions

CHAPTER 3: Random Variables and Probability Distributions

Concept of a Random Variable: 3.1

? The outcome of a random experiment need not be a number.

? However, we are usually interested not in the outcome itself, but rather in some measurement of the

outcome.

Example: Consider the experiment in which batteries coming off an assembly line were examined until a

good one (S) was obtained.

S = {S, FS, FFS, . . .}.

We may be interested in the number of batteries examined before the experiment terminates.

A random variable is a function that associate a real number with each element in the sample space.

Example: Tossing two coins

S = {HH, TT, HT, TH}

Let X = # of heads observed.

Example: A group of 4 components is known to contain 2 defectives. An inspector tests the components

one at the time until the 2 defectives are located. Let X denote the number of the test on which the second

defective is found.

Two types of random variables

? A discrete random variable is a random variable whose possible values either constitute a finite set

or else can be listed in an infinite sequence.

? A random variable is continuous if its set of possible values consists of an entire interval on the

number line.

Many random variables, such as weight of an item, length of life of a motor etc., can assume any value

in certain intervals.

1

Discrete Probability Distributions: 3.2

Probability mass function of a discrete random variable X is defined by

f (x) = P (X = x)

Example: tossing two coins

X = # of heads.

f (0) = P (X = 0) = P (TT) = 1/4

f (1) = P (X = 1) = P (HT, TH) = 1/2

f (2) = P (X = 2) = P (HH) = 1/4

Example: An information source produces symbols at random from a five-letter alphabet:

S = {a, b, c, d, e}.

The probabilities of the symbols are

p(a) =

1

1

1

1

, p(b) = , p(c) = , p(d) = p(e) =

.

2

4

8

16

A data compression system encodes the letters into binary strings as follows:

a

b

c

d

e

1

01

001

0001

0000

Let the random variable Y be equal to the length of the binary string output by the system.

f (1) = P (Y = 1) =

f (2) = p(Y = 2) =

f (3) = p(Y = 3) =

f (4) = p(Y = 4) =

f (x) = P (X = x) satisfies the following conditions:

1. fP(x) ¡Ý 0

2.

f (x) = 1

Example: A box contains 5 balls numbered 1, 2, 3, 4, and 5. Three balls are drawn at random and without

replacement from the box. If X is the median of the numbers on the 3 chosen balls, then what is the

probability function for X, where nonzero?

Solution

2

Example: Determine c so that the function f (x) can serve as the probability mass function of a random

variable X:

f (x) = cx for x = 1, 2, 3, 4, 5

Solution:

The cumulative distribution function: F (x) of a discrete random variable X with probability mass function

f (x) is defined for every number x by

X

f (t)

F (x) = P (X ¡Ü x) =

t¡Üx

Example: Assume that

f (2) = p(X = 2) = 1/6 f (3) = p(X = 3) = 1/3

f (4) = p(X = 4) = 1/2

Then

F (2) =

F (3) =

F (4) =

F (x) =

Example: A mail order computer business has six telephone lines. Let X denote the number of lines in use

at a specified time. Suppose that the probability mass function of X is given below

x

p(x)

0

0.10

1

0.15

2

0.20

3

0.25

4

0.20

5

0.06

6

0.04

a. Find the cumulative distribution function

b. Find the probability that

{at most 3 lines are in use}.

c. Find the probability that

{at least 4 lines are in use}.

3

Example:

If X has the cumulative distribution function:

find the probability mass function.

?

0

?

?

?

?

? 1/3

1/2

F (x) =

?

?

? 5/6

?

?

1

if

if

if

if

if

x ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download