chapter 4 discrete random variables. two types of random variables random variable –variable that...

Chapter 4

Discrete Random Variables

Two Types of Random Variables

Random Variable– Variable that assumes numerical values

associated with random outcomes of an experiment

– Only one numerical value is assigned to each sample point

Two types of Random Variable– Discrete– Continuous

Two Types of Random Variables

• Discrete Random Variable– Random variable that has a finite, or

countable number of distinct possible values– Example – number of people born in July

• Continuous Random Variable– Random variable that has an infinite number

of distinct possible values– Average age of people born in July

Probability Distributions for Discrete Random Variables

Two requirements that must be satisfied:

1

0

xp

xp for all values of x

Where the summation of p(x) is over all possible values of x

1.

2.


Experiment - tossing 2 coins simultaneously

Random variable X – number of heads observed

X can assume values of 0, 1 and 2Calculate the probability associated with each value of X

Probability Distribution for Coin-Toss Experiment

X p(x)

0 ¼

1 ½

2 ¼


Probability Distribution of Discrete Random Variable X – Other forms

412

2141411

410

HHPxP

HTPTHPxP

TTPxP

Expected Values of Discrete Random Variables

The mean, or expected value of a discrete random variable is:

xxpxExpected Value of x (number of heads observed)

X P(x) Xp(x)

0 ¼ 0

1 ½ ½

2 ¼ ½

Expected Value 1

Expected Values of Discrete Random Variables

The variance of a discrete random variable is:

and standard deviation is

21411221114110

222

222

xpxx

212

The Binomial Random Variable

Binomial Random variable – An experiment of n identical trials– 2 possible outcomes on each trial, denoted as

S (success) and F (failure)– Probability of success (p) is constant from trial

to trial. Probability of failure (q) is 1-p– Trials are independent– Binomial random variable – number of S’s in n

trials


Computer retailer selling desktop (D) and laptop (L) PCs online. Sales of 80% desktop, 20% laptop.

What is the probability that next 4 sales are Laptops?

Sample points for next 4 online purchasesDDDD LDDD

DLDD

DDLD

DDDL

LLDD

LDLD

LDDL

DLLD

DLDL

DDLL

DLLL

LDLL

LLDL

LLLD

LLLL


Use multiplicative rule to calculate probabilities of the possible outcomes

P(DDDD) = .8*.8*.8*.8=.84=.4096

P(LDDD) = .2*.8*.8*.8=.2*.83=.1024

…..

P(LLLL) = .2*.2*.2*.2=.24=.0016


What is the probability that 3 of the next 4 online sales are laptops?P(3 of the next 4 customers purchase laptops) = 4(.2)3(.8)=4(.0064) = .0256What is the probability that 3 of the next 4 online sales are desktops?P(3 of the next 4 customers purchase desktops) = 4(.8)3(.2)=4(.1024) = .4096Do you see a pattern?


Formula for the probability distribution p(x)

Where p = probability of success on single trial

q = 1-p

n = Number of trials

x = number of successes in n trials

xnxqpx

nxp

)(

)!(!

!

xnx

n

x

n


P(3 of the next 4 customers purchase laptops) = 4(.2)3(.8)=4(.0064) = .0256

x=3, n=4

)8(.)2(.4)8(.)2(.321

4321

)8(.)2(.3

43

33

343

xnxqpx

nxp


P(3 of the next 4 customers purchase desktops) = 4(.8)3(.2)=4(.1024) = .4096x=3, n=4

)2(.)8(.4)2(.)8(.321

4321

)2(.)8(.3

43

33

343

xnxqpx

nxp


Mean:

Variance:

Standard deviation

np

npq2

npq


Using Binomial Tables

Binomial tables are cumulative tables, entries represent cumulative binomial probabilities

Make use of additive and complementary properties to calculate probabilities of individual x’s, or x being greater than a particular value.


If x < 2, and p =.2, n =10, then P(x<2) =.678

If x = 2, and p =.2, n =10, then P(x=2) = P(x<2) - P(x<1)=.678-.376 = .302

If x >2, and p = .2, n =10, then P(x>2) = 1- P(x<2) =1-.678 = .322

.30.20.10.05.01

.383.678.930.9881.0002

.149.376.736.914.9961

.650.879.987.9991.0003

4

0

.850.967.9981.0001.000

.028.107.349.599.904

Binomial probabilities for n=10 (partial table)

pk

chapter 4 discrete random variables. two types of random variables random variable –variable that...

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