Random Variables

Learn how to characterize the pattern of the distribution of values that a random variable may have, and how to use the pattern to find probabilities

Budhi SetiawanTeknik Sipil - UNSRI

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What is a Random Variable?

Random Variable: an outcome or event may be identified through the value(s) of a function, which usually denoted with a capital letter

Two different broad classes of random variables:

1. A continuous random variable can take any value in an interval or collection of intervals.

2. A discrete random variable can take one of a countable list of distinct values.

If the value of X represent flood above mean level, then X > 7 meter stand for the occurrence of floods above 7 meter

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Random factors that will determine how enjoyable the event is:

Temperature: continuous random variable (any value, integer or decimal)

Number of airplanes that fly overhead: discrete random variable (integer only)

Example: Random Variables at an Outdoor Graduation or Wedding

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• What is the probability that three tosses of a fair coin will result in three heads?

• Assuming boys and girls are equally likely, what is the probability that 3 births will result in 3 girls?

• Assuming probability is 1/2 that a randomly selected individual will be taller than median height of a population, what is the probability that 3 randomly selected individuals will all be taller than the median?

Example: Random Variables:Probability an Event Occurs 3 Times in 3 Tries

Answer to all three questions = 1/8.

Discrete Random Variable X = number of times the “outcome of interest” occurs in three independent tries.

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Discrete Random Variables

Discrete random variable: can only result in a countable set of possibilities –

often a finite number of outcomes, but can be infinite.

X the random variable.

k = a number the discrete r.v. could assume.

P(X = k) is the probability that X equals k.

Example: It’s Possible to Toss Forever

Repeatedly tossing a fair coin, and define:X = number of tosses until the first head occurs

Any number of flips is a possible outcome.

P(X = k) = (1/2)k

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Probability Distribution of a Discrete R.V.

Using the sample space to find probabilities:

Step 1: List all simple events in sample space.

Step 2: Find probability for each simple event.

Step 3: List possible values for random variable X and identify the value for each simple event.

Step 4: Find all simple events for which X = k, for each possible value k.

Step 5: P(X = k) is the sum of the probabilities for all simple events for which X = k.

Probability distribution function (pdf) X is a table or rule that assigns probabilities to possible values of X.

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Example:How Many Girls are Likely?

Family has 3 children. Probability of a girl is ?What are the probabilities of having 0, 1, 2, or 3 girls?

Sample Space: For each birth, write either B or G. There are eight possible arrangements of B and G for three births. These are the simple events.

Sample Space and Probabilities: The eight simple events are equally likely.

Random Variable X: number of girls in three births. For each simple event, the value of X is the number of G’s listed.

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Example: How Many Girls? (cont)

Probability distribution function for Number of Girls X:

Value of X for each simple event:

Graph of the pdf of X:

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Conditions for Probabilities for Discrete Random Variables

Condition 1

The sum of the probabilities over all possible values of a discrete random variable must equal 1.

Condition 2

The probability of any specific outcome for a discrete random

variable must be between 0 and 1.

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Cumulative Distribution Function of a Discrete Random Variable

Cumulative distribution function (cdf) for a random variable X is a rule or table that provides the probabilities P(X ≤ k) for any real number k.

Cumulative probability = probability that X is less than or equal to a particular value.

Example: Cumulative Distribution Function for the Number of Girls (cont)

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Finding Probabilities for Complex Events

Example: A Mixture of Children

pdf for Number of Girls X:

What is the probability that a family with 3 children will have at least one child of each sex?

If X = Number of Girls then either family has one girl and two boys (X = 1) or two girls and one boy (X = 2).

P(X = 1 or X = 2) = P(X = 1) + P(X = 2) = 3/8 + 3/8 = 6/8 = 3/4

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Expectations for Random Variables

The expected value of a random variable is the mean value of the variable X in the sample space, or population, of possible outcomes.

If X is a random variable with possible values x1, x2, x3, . . . , occurring with probabilities p1, p2, p3, . . . , then the expected value of X is calculated as

ii pxXE

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Standard Deviation for a Discrete Random Variable

The standard deviation of a random variable is essentially the average distance the random variable falls from its mean over the long run.

If X is a random variable with possible values x1, x2, x3, . . . , occurring with probabilities p1, p2, p3, . . . , and expected value E(X) = , then

ii

ii

pxX

pxXVX

2

22

ofDeviation Standard

of Variance

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Binomial Random Variables

1. There are n “trials” where n is determined in advance and is not a random value.

2. Two possible outcomes on each trial, called “success” and “failure” and denoted S and F.

3. Outcomes are independent from one trial to the next.

4. Probability of a “success”, denoted by p, remains same from one trial to the next. Probability of “failure” is 1 – p.

Class of discrete random variables = Binomial -- results from a binomial experiment.

Conditions for a binomial experiment:

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Examples of Binomial Random Variables

A binomial random variable is defined as X=number

of successes in the n trials of a binomial experiment.

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Finding Binomial Probabilities

p = probability win = 0.2; plays of game are independent.

X = number of wins in three plays.

What is P(X = 2)?

knk ppknk

nkXP

1

!!

!for k = 0, 1, 2, …, n

Example: Probability of Two Wins in Three Plays

096.0)8(.)2(.3

2.12.!23!2

!32

12

232

XP

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Binomial Probability Distribution

Binomial distribution is based on events in which there are only two possible outcomes on each occurrence.

Example: Flip a coin 3 times the possible outcomes are (heads = hits; tails =

misses):

HHH, HHT, HTT, TTT, TTH, THH, THT, AND HTH

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Binomial Probability Distribution

Example: Flip a coin 3 times the possible outcomes are (call heads = hits; tails = misses):

Possible Outcomes of Coin Flipped 3 times

Outcome No. Hits (x)

HHH

HHT

THH

HTH

HTT

THT

TTH

TTT

3

2

2

2

1

1

1

0

Frequency Dist of data

X f

3

2

1

0

1

3

3

1

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Binomial Probability Distribution

0

0.5

1

1.5

2

2.5

3

Frequency

Frequency Distribution

0 1 2 3

HITS

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Probability Associated with Hits

Hits Frequency Probability

3

2

1

0

1

3

3

1

.125

.375

.375

.125

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Binomial Probability Distribution

0

0.5

1

1.5

2

2.5

3

Frequency

Frequency Distribution

0 1 2 3

HITS

.125

.250

.375

.500

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Binomial Probability DistributionBinomial Probability Distribution

The preceding bar graph is symmetrical; this will always be true for the binomial distribution when p= 0.5.

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Expected Value and Standard Deviation for a Binomial Random Variable

For a binomial random variable X based on n trials and success probability p,

pnp

npXE

1 deviation Standard

Mean

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Example: Extraterrestrial Life?

50% of large population would say “yes” if asked, “Do you believe there is extraterrestrial life?”

Sample of n = 100 is taken.

X = number in the sample who say “yes” is approximately a binomial random variable.

In repeated samples of n=100, on average 50 people would say “yes”. The amount by which that number would differ from

sample to sample is about 5.

55.)5(.100 deviation Standard

50)5(.100 Mean

XE