CHAPTER 3: Random Variables and Probability Distributions

[Pages:16]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.

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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 2

,

p(b)

=

1 4

,

p(c)

=

1 8

,

p(d)

=

p(e)

=

1 16

.

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

a1 b 01 c 001 d 0001 e 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. f (x) 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

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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

F (x) = P (X x) = f (t)

tx

Example: Assume that

Then F (2) = F (3) = F (4) = F (x) =

f (2) = p(X = 2) = 1/6 f (3) = p(X = 3) = 1/3 f (4) = p(X = 4) = 1/2

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

0

1

2

3

4

5

6

p(x) 0.10 0.15 0.20 0.25 0.20 0.06 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}.

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Example: If X has the cumulative distribution function:

find the probability mass function.

0

1/3

F (x) = 1/2

5/6

1

if x < 1 if 1 x < 4 if 4 x < 6 if 6 x < 10 if x 10,

Solution:

Continuous Probability Distribution: 3.3

A density curve is a curve that ? is always on or above the horizontal axis, and ? has area exactly 1 underneath it.

A density curve describes the overall pattern of a distribution. The area under the curve and above any range of values is the proportion of all observations that fall in that range.

Definition 3.6 from the book: The function f (x) is a probability density function for the continuous random variable X, defined over the set of real numbers R, if

1. f (x) 0, for all x R.

2.

-

f (x)dx

=

1

3. P (a < X < b) =

b a

f

(x)dx

Ex. 9 on p. 73: The proportion of people who respond to a certain mail-order solicitation is a continuous random variable X that has density function

f (x) =

2(x+2) 5

if 0 < x < 1

0

otherwise,

(a)Show that P (0 < X < 1) = 1

(b) Find the probability that more than 1/4 but fewer than 1/2 of the people contacted will respond to this type of solicitation.

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density(x)$y 0.0 0.2 0.4 0.6 0.8 1.0 1.2

?? ?

? ?

? ?

?

?

?

?

? ? ? ? ? ???????? ? ? ??????

?

?

?

? ? ? ? ? ? ? ? ? ? ?????

0.5

1.0

1.5 density(x)$x

2.0

2.5

3.0

Figure 1: Density histogram for the data in Ex 9 on p. 21

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Example The mileage (in thousands of miles) that car owners get with a certain kind of tire is a random variable having the density function

f (x) =

1 20

e-x/20

if x > 0

0

if x 0,

Find the probabilities that one of these tires will last (a) at most 10, 000 miles;

(b) anywhere from 16, 000 to 24, 000 miles;

(c) at least 30, 000 miles.

Solution:

Example: The pdf of the samples of speech waveforms is found to decay exponentially at a rate , so the following pdf is proposed:

f (x) = ce-|x| - < x < Find the constant C, and then find the probability P [|X| < ].

Solution

Example Suppose the reaction temperature X in a certain process has a uniform distribution

f (x) =

1 10

,

if -5 x 5,

0 otherwise,

Compute P (X < 0), P (-2 < X < 3), and P (-7 < X < 1)

Solution:

Example: The cumulative distribution function of checkout time duration X is

0

F (x) =

x2 4

1

if x < 0,

if 0 x < 2, if x 2

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(a) Use this to compute P (X 1) and P (0.5 X 1) (b) Find the density function of X Solution

Review: 1. A sale engineer for a manufacturer of high-speed grinding equipment has just returned from visiting five possible clients. She believes that the following table describes the distribution of the number of sales she will make:

x(#of sales) 0 1 2 3 4 5 P (X = x) 0.05 0.3 0.3 0.2 0.1 0.05 a. Compute and plot the cumulative function F (x) = P (X x). b. Find the probability that the sales engineer will make more than 3 sales. c. Find the probability that she makes at least 2 sales.

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2. The output of a chemical process is continually monitored to ensure that the concentration remains within acceptable limits. Whenever the concentration drifts outside the limits, the process is shut down and re-calibrated. Let X be the number of times in a given week that the process is calibrated. Let X be the of times in a given week that the process is calibrated. The following is the cumulative function F (X),

0

0.17

F (x)

=

0.53 0.84

0.97

1

if x < 0 if 0 x < 1 if 1 x < 2 if 2 x < 3 if 3 x < 4 if x 4

a. What is the probability that the process is re-calibrated fewer than two times during a week. b. What is the probability that the process is re-calibrated more than three times during a week? c. What is the probability that the process is re-calibrated exactly once during a week? d. What is the probability that the process is not re-calibrated at all during a week? e. What is the most probable number of re-calibrations to occur during a week?

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