Business Statistics

Lecture 5: Random Variables and Discrete Probability Distributions

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Agenda

Discrete probability distributions

random variables and probability distributions

expected value and variance (discrete random variable)

some specific discrete probability distributions: binomial distribution and Poisson distribution

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Random variables

Random experiment:

a process that produces an outcome

actual outcome is uncertain, but whose possible outcomes are known in advance

Random variable:

A variable whose numerical value is determined by the outcome of a random experiment.

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Discrete and continuous random variables

Random variables may be either discrete or continuous a random variable is discrete if it can assume only a finite or

countable number of values

a random variable is continuous if it can assume an uncountable (infinite) number of values

0 1 1/2 1/4 1/16 0 1 2 3 ...

Therefore, the number of values is countable

Therefore, the number of values is uncountable

After the first value is defined, the second value and any value thereafter are known

After the first value is defined, any number can be the next one

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Probability distribution A probability distribution is a distribution of the

probabilities associated with each of the values that a random variable can take

Each random variable has a probability mass function (discrete) or probability density function (PDF: continuous) associated with it, describing the probability distribution of the random variable

Probability distributions

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Continuous Probability Distributions

Binomial (이항)

Poisson

(포아송)

Probability Distributions

Discrete Probability Distributions

Normal

Exponential

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1)(.2

allfor1)(0 .1

al l

ix

i

ii

xp

xxp

A (discrete) probability distribution can be used to calculate probabilities of different events

e.g.,

4

3

4

1

2

1)2()1()21( XPXPXP

Requirements of discrete probability distr.

If a discrete random variable can take values xi, then the following must be true:

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Expected value and variance Researchers usually have an interest in specific

characteristics of the outcomes of a random variable, e.g. the average of all possible (future) outcomes

They can not calculate these characteristics, but they may form expectations about them

Two specific characteristics (parameters) of a random variable: expected value variance

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ix

ii xpxXEa l l

)()(

weighted average of the possible values of X, where the weights are the corresponding probabilities of each xi

interpretation: long-run average if random experiment is repeated a large number of times

Expected value – definition Expected value:

given a discrete random variable X with values xi, that occur with probabilities p(xi), the expected value of X is

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ii xpxXEa l l

222)(

weighted average of the squared deviations of the possible values of X from the expected value , where the weights are the corresponding probabilities

Standard Deviation () is the square root of the variance (2)

Variance – definition Variance:

given a discrete random variable X with values xi, that occur with probabilities p(xi), and with expected value E(X) = , the variance V(X) = 2 of X is

2

al l

2222 )(

:ncalculatioShortcut

ix

ii xpxXE

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Discrete probability distributions

Following discrete probability distributions and their properties will hereafter be explained: Binomial distribution

Poisson distribution

Insight into random experiment determines appropriate probability function (e.g. the probability distribution for the tossing experiment)

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Binomial distribution

Used when interested in the number of favorite or successful outcomes in a series with a given number of repeated trials

Examples: number of defective items in a series of 200 produced number of ‘heads’ when flipping a coin 3 times number of even numbers when rolling a die 10 times

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Binomial distribution – Bernoulli experiment

Binomial (Bernoulli) experiment: a series of n independent trials with only two possible outcomes, and the same probability of ‘success’ (and ‘failure’)

there are n trials (n is finite and fixed)

all trials are independent

each trial results either in ‘success’ or in ‘failure’

the probability of success (p) is the same in all trials

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Binomial distribution – definition

A binomial random variable (X) counts the number of successful outcomes in a binomial (Bernoulli) experiment

Notation: X ~ Bin(n, p)

By definition X is a discrete random variable

By definition X has possible outcomes x = 0, 1, …, n

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1!0 and 1)1(! w ith

t)coefficien (binom ial )!(!

!where

nnn

xnx

nC n

x

Binomial distribution – expression

In general, the binomial probability is calculated by the following probability function:

1n xn x

xP X x p x C p p

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Binomial distribution – application

conditions required for a binomial experiment are met:

converter can be either defective (‘success’) or good (‘failure’)

there is a fixed number of trials (n = 3)

we assume that the converter state is independent on one another

the probability of a converter being defective does not change from converter to converter (p = .05)

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x P(X) 0 .8574 1 .1354 2 .0071 3 .0001

Binomial distribution – application

Solution (cont.): define a ‘success’ as ‘item is defective’

define X = number of defective items in sample

then X ~ Bin(n = 3, p = .05)

hence 0 3 0

1 3 1

2 3 2

3 3 3

3!( 0) (0) (.05) (.95) .8574

0!(3 0)!

3!( 1) (1) (.05) (.95) .1354

1!(3 1)!

3!( 2) (2) (.05) (.95) .0071

2!(3 2)!

3!( 3) (3) (.05) (.95) .0001

3!(3 3)!

P X p

P X p

P X p

P X p

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Mean and variance of a binomial rand. var.

)1()( a n d )( 2 pn pXVn pXE

If X ~ Bin(n, p), then the mean and variance of X are found as

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Poisson distribution

Used when interested in the number of occurrences in an interval (e.g., a fixed amount of time or within a specified region)

Examples:

number of customers entering a shop per hour

number of defects in a production process per hour

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Poisson distribution – experiment

Properties of the Poisson experiment:

the number of successes (events) that occur in any interval is independent of the number of successes that occur in any other interval

the probability that a success will occur in an interval

is the same for all intervals of equal size

varies with the size of the interval [the average number of successes is proportional to the size of the interval]

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Poisson distribution – definition

A random variable (X) counts the number of successes that occur during a given interval (of time, length, area, etc.) in a Poisson experiment

Notation: X ~ Poisson(), where is the average number of successes in this interval

By definition X is a discrete random variable

By definition X has possible outcomes x = 0, 1, …

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( ) 0,1,2,!

xeP X x p x x

x

Poisson distribution – expression

According to the Poisson probability distribution, the probability of x occurrences in an interval is:

If X ~ Poisson(), then both the mean and the variance of X are:

where is the mean number of occurrences in that interval & the value of e = 2.71828…, the base of the natural logarithm

) ( and ) ( X V X E

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1 2 11

2 (2) .18392! 2

e eP X p

Poisson distribution – calculation

0

0.1

0.2

0.3

0.4

1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5

Poisson probability function with = 1:

1 0

110 (0) .3679

0!

eP X p e

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Example Example

cars arrive at a tollbooth at a rate of 360 cars per hour

what is the probability that exactly two cars will arrive during a specified one-minute period? (use formula)

Solution: let X denote the number of arrivals per one-

minute period the mean number of arrivals per minute is µ =

360/60 = 6 cars it follows that X ~ Poisson(6)

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6 26

2 (2) .04462!

eP X p

Cont’d

Example (cont.):

since X ~ Poisson(6) the desired probability can be found as

follows