Lecture 06. Discrete Random Variables

7/28/2019 Lecture 06. Discrete Random Variables

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Statistics

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ST 361: Statistics for EngineersDiscrete Random Variables

Kimberly Weems

[email protected]

5260 SAS Hall


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Statistics

Random Variables

Motivating Ex.: Assume that all mens basketball teams playingthis season are equally strong. We are interested in the number

of points scored by NC State in each game.

Before each game, we know the population of possible values.

Each value occurs with some probability. However, we do not know what will be the number of points

scored by NC State during the next game.

The outcome is random, hence a random variable.


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Random Variables

Ex: in an experiment to measure the speed of light, theinaccuracies of the measurement process make the potential

population of measurements infinite, yet the observer must settle

for a finite sample of measurements.


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Random Variables

Ex: The possible outcomes are 1,2,3,4,5,and 6. The outcomes

can be equally likely (die is perfect cube), but cannot say

what will come up. As before, the outcome is random.

A variable that associates a number with the outcome of a

random experiment is called a random variable.

Formal defn: A random variable (rv) = a function that assigns

a real number to each outcome in the sample space of a

random experiment.

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Continuous & Discrete Random Variables

A discrete random variable is a random variable with a finite(or countably infinite) range. They are usually integer counts,

e.g., number of errors or number of bit errors per 100,000

transmitted (rate).

A continuous random variable is a random variable with an

interval (either finite or infinite) of real numbers for its range.

Its precision depends on the measuring instrument.

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Example: Voice Lines

A voice communication system for a business contains 48external lines. At a particular time, the system is observed,

and some of the lines are being used.

LetXdenote the number of lines in use.Then,Xcan assume any of the integer values 0 through 48.

The system is observed at a random point in time. If 10 lines

are in use, thenx = 10.

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Probability Distributions

Recall: A random variableXassociates the outcomes of a

random experiment to a number on the number line.

The probability distribution of the random variableXis a

description of the probabilities with the possible numerical

values ofX.

A probability distribution of a discrete random variable can be:

1. A list of the possible values along with their probabilities.

2. A formula that is used to calculate the probability inresponse to an input of the random variables value.

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Probability mass function

The probability distribution of a discrete random variable is

called a probability mass function (pmf).

Gives as a list of values along with their probabilities:

Representation (for discrete with finite number of values):

Value of X x1 x2 .. . xn

Probability p(x1) p(x2) . . p(xn)


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Probability distribution of a discrete

random variable

A Probability Mass Function satisfies

For all valuesx:

We have:

To give a probability mass function, specify the values

and their correspondingprobabilities.

( ) ( )p x P X x

0 ( ) 1p x

( ) 1x

p x


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Cumulative Distribution Function

The cumulative distribution function is built from the probability

mass function (and vice versa).

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The cumulative distribution function of a discrete random variable ,

denoted as ( ), is:

(1)

(2) 0 1

(3) If , then

i

i

x x

X

F x

F x P X x p x

F x

x y F x F y


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Summary Numbers of a Probability

Distribution The mean is a measure of the centerof a probability

distribution.

The variance is a measure of the dispersion or variability of a

probability distribution.

The standard deviation is anothermeasure of the dispersion. It

is the positive square root of the variance.

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Mean Defined

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The or of the discrete random variable X,denoted as or

mean expected, isvalue

x

E X

E X x p x

The mean is the weighted average of the possible values ofX,

the weight of each valuex represents how likely the occurrence

of value x is. It represents the center of the distribution. It is

also called the arithmetic mean.

Ex. Ifp(x) is the pmf representing the loading on a long, thin

beam, thenE(X) is the fulcrum or point of balance for the beam.


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Variance Defined

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2

2 22 2 2

The of X, denoted as or , isvariance

x x

V X

V X E X x p x x p x

The variance is the measure of dispersion or scatter in the

possible values forX.It is the average of the squared deviations from the distribution

mean.


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Variance Defined

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The mean is the balance point. Distributions (a) & (b) have

equal mean, but (a) has a larger variance.


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Variance Formula Derivations

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2

2 2

2 2

2 2 2

2 2

is the formula

2

2

2

is the form

definitional

computatio ull ana

x

x

x x

x

x

V X x p x

x x p x

x p x xp x p x

x p x

x p x

The computational formula is easier to calculate manually.


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Ex. Digital Channel

There is a chance that a bit transmitted through a digital transmission

channel is an error. Xis the number of bits received in error of the next 4transmitted. Use table to calculate the mean & variance.

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x p(x) x*p(x) (x-0.4)2 (x-0.4)2*p(x) x2*p(x)

0 0.6561 0.0000 0.160 0.1050 0.0000

1 0.2916 0.2916 0.360 0.1050 0.2916

2 0.0486 0.0972 2.560 0.1244 0.1944

3 0.0036 0.0108 6.760 0.0243 0.0324

4 0.0001 0.0004 12.960 0.0013 0.0016Totals = 0.4000 0.3600 0.5200

= Mean = Variance (2) = E(x

2)

= 2

= E(x2) -

2= 0.3600

Definitional formula

Computational formula


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Particular class of Discrete Random

Variables1. Flip a coin 10 times. X= # heads obtained.

2. A worn tool produces 1% defective parts. X= # defective

parts in the next 25 parts produced.

3. A multiple-choice test contains 10 questions, each with 4

choices, and you guess. X= # of correct answers.

4. Of the next 20 births, letX= # females.

5. Assume your favorite basketball team plays 10 games (of

equal difficulty). Let X = the number of times that it wins the

game.


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Binomial Random Variables

These examples are binomial experiments having the following

characteristics:

1. Fixed number of trials (n).

2. Each trial is termed a success (S) or failure (F). Xis the # of

successes.

3. The probability of success in each trial is constant (p).

4. The outcomes of successive trials are independent.


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Binomial distribution

Commonly used discrete distribution

Bernoulli trial [definition]

Single trial with only 2 possible outcome (S or F)

The probability of success is p.

Binomial variable

Considern Bernoulli independent trials

Each trial has the same probability of successp

Xcounts the number of successes in these ntrials.

( , )X Bin n p

( )X Bernoulli p


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The probability mass function of X is:

Lets calculate pk=Pr(X=k), and k=0,1,n[Recall that X counts the number of successes in n trials ]

1) probability of an outcome comprised of k successes and (n-k)

failures

.

Value of X 0 1 .. . n

Probability p0 p1 . . pn

, , ..., , , ...,

k n k

S S S F F F


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The probability mass function of X is:

Lets calculate pk=Pr(X=k), and k=0,1,n[Recall that X counts the number of successes in n trials ]

1) probability of an outcome comprised of k success and (n-k)

failures

2) the number of such outcomes:

.

Value of X 0 1 .. . n

Probability p0 p1 . . pn

(1 )k n kp p

!

!( )!

nn

kk n k


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Lets calculate pk=Pr(X=k), and k=0,1,n

[Recall that X counts the number of successes in n trials ]

1) probability of an outcome comprised of k success and (n-k)

failures

2) the number of such outcomes:

The Binomial PMF is:

(1 )k n kp p

!!( )!

nnkk n k

( ) (1 )k n kk np P X k p pk


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Factorials

0!=1

1!=1

2!=1X2=2

3!=1X2X3=6

4!=1X2X3X4=245!=1X2X3X4X5=120

General formula :

[this formula is helpful for canceling]

! ( 1)!n n n


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Example 3-16: Digital Channel

The chance that a bit transmitted through a digital transmission

channel is received in error is 0.1. Assume that the transmission

trials are independent. LetX= the number of bits in error in the

next 4 bits transmitted. FindP(X=2).


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Example 3-16: Digital Channel

Let E denote a bit in error

Let O denote an OK bit.

Sample space &x listed in table.

6 outcomes wherex = 2.

Prob of each is 0.12*0.92 = 0.0081

P(X=2) = 6*0.0081 = 0.0486

Outcome x Outcome x

OOOO 0 EOOO 1

OOOE 1 EOOE 2

OOEO 1 EOEO 2

OOEE 2 EOEE 3

OEOO 1 EEOO 2

OEOE 2 EEOE 3

OEEO 2 EEEO 3

OEEE 3 EEEE 4 2 24

2 0.1 0.92

P X


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Mean/Variance of the Binomial RV

IfXhas aBin(n,p) distribution, the probability mass functionofXis given by

fork = 0,1,2,,n

The Mean and Variance ofXare:

X = np , and 2

X = np(1-p)

Example: X has Bin(5, 0.6) distribution.

Then the mean is X = 50.6 = 3 ;

the variance is 2X = 5 0.60.4 = 1.2

1n kk

nP X k p p

k

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Lecture 06. Discrete Random Variables