RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

UNIT 4 - PROBABILITY

DISTRIBUTIONS

■ Random Variables & Probability Distributions ■ Binomial Distribution

RANDOM VARIABLES AND

PROBABILITY

DISTRIBUTIONS

■ Random Variables

■ Probability Distributions

Random Variables - Definition

⚫

A

random variable

is a variable whose value is

determined by the outcome of a random experiment.

⚫

Typically we use capital letters to represent random

variables, and lower case letters to represent

particular values they can take

■ Eg : The random variable X represents the face of an unbiased

Random Variables

• There are two types of random variables

• Discrete Random Variables – can assume only countable values

• Examples

• Number of chairs

• Number of lotto winners

• Continuous Random Variables – can assume any value in an interval

• Examples

• weight

• height

Probability Distributions

• A probability distribution is:

• A mathematical function that assigns probabilities to the values of a random variable

• Eg:

• A listing of all the outcomes of an experiment and the probability of each outcome

• Eg:

• A function which describes how probabilities are distributed over the values of a random variable

• Eg: We roll two dice. Let the random variable X = “sum of the

two faces”

X 0 1 2

Properties of Probability

Distributions

•

Each probability MUST be greater than or equal

to zero

• f(x) ≥ 0 or P(X = x) ≥ 0

• ie. Probabilities cannot be negative

•

The sum of ALL probabilities MUST be equal to 1

Discrete vs. Continuous Probability

Distributions

• The discrete probability distribution lists all

possible values than the variable can assume and corresponding probabilities.

• For a continuous random variable, it is

Example – Discrete Probability

Distributions

⚫ Example 1:

⚫ If we roll two dice and let the random variable X be the sum of the values facing up

a. Calculate the probability distribution of X b. Determine the probability of X = 5

c. Determine the probability of the sum of the

Distributions

•Example 2:

•Let

•be the probability distribution of X.

Summarising Probability Distributions

• Probability distributions of random variables are

summarized in various ways, using the key ideas of centre, spread, and shape.

• The most important measures are:

• Centre — the mean or expected value (”expectation”)

• Spread — the variance and the standard deviation

Expected Value

•

The

expected value

of a random variable is

simply the mean of a random variable

•

It is denoted as E(X) – read ‘the expectation of

X’

•

Calculating the expected value:

• Formula: Σ x P(X) or Σ x f(x)

• Translation: multiply each value of x by its

Variance

• The variance of a random variable measures

the spread of the distribution

• It is denoted by V(X) or Var(X) • Calculating V(X):

• Formula: Σ x2 P(X) – [Σ x P(X)] 2

• Translation:

●multiply the square of each x value by its corresponding

probability and sum the results

Example – Expected Value & Variance

• Example 1:

• Let x be the number of magazines a person reads every

week. Based on a sample survey of adults, the following probability distribution table was prepared:

• Calculate the mean and standard deviation.

x 0 1 2 3 4 5

Example – Expected Value & Variance

• Example 2:

• Loraine Corporation is planning to market a new

makeup product. According to the analysis made by the financial department of the company, it will earn an annual profit of $4.5 million if this product has high sales, an annual profit of $1.2 million if the sales are mediocre and it will lose $2.3 million if the sales are low. The probabilities of these

three scenarios are 0.32, 0.51 and 0.17

• Calculate the mean and standard deviation of the

BINOMIAL DISTRIBUTION

(DISCRETE PROBABILITY

DISTRIBUTION)

■ What is the Binomial Distribution (BD)?

The Binomial Distribution

• One of the most widely used Discrete Probability

Distributions

• Applied to find the probability that an event occurs

x times in n performances of an experiment.

• To apply the Binomial distribution the random

The Binomial Distribution

Binomial Distribution – Example I

•

Individuals with a certain gene have a 0.70

probability of eventually contracting a certain

disease. What is the probability that 5 out of

10 randomly selected persons with the certain

will have the disease?

•

This is a Binomial Distribution. Why?

• 10 identical trials (testing for the disease)

• Each trial has only 2 possible outcomes (have the disease, does not have the disease)

Binomial Distribution – Example II

•

Suppose a survey in a particular country

indicated that 9 out of 10 cars carry liability

insurance. If four cars in that country are

involved in accidents what is the probability

that …

•

This is a Binomial Distribution. Why?

• 4 identical trials (4 cars in accidents)

• Each trial has only 2 possible outcomes (have liability insurance, does not have liability insurance)

• We assume a constant probability (9 out of 10 – 9/10 or 0.9)

Binomial Distribution

•

The formula used to calculate the probability

for the binomial distribution is:

•

Where :

• x = probability of interest • p = probability of success • n = number of trials

• q = 1 – p (probability of failure) • n-x = number of failures

x

•

This is calculated

using:

n denotes the total number of

elements

x denotes the number of elements selected per selection

What is

n

C

_x

? - II

• n! – ‘n factorial’

• n! = n x (n-1) x (n-2) x (n-3) x . . . . x 3 x 2 x 1

• Example 1: 4! = 4 x 3 x 2 x 1 = 24

Playing Lotto - I

• Write down any six numbers between 0 & 36

inclusive

• Watch the draw . . ..

• http://www.youtube.com/watch?v=7fdBLzoIOjs&feature=r

Playing Lotto - II

•

Calculate the total number of combinations of

any 6 numbers between 0 and 36

•

It would be:

•

To ensure you win the lotto on a certain day

you would have to buy every possible

combination – ie 2,324,784 tickets.

Binomial Distribution

⚫ NB Formula:

⚫ Step 1

: Calculate

C

⚫ Step 2

: Calculate

p

⚫ Step 3

: Calculate

q

Back to the Binomial Distribution

• Reminders:

• Only 2 outcomes: (i) success or (ii) failure

• Known probability of success

• Probability Distribution Function:

• n = number of trials

• p = probability of success

Example I - Binomial Distribution

• Suppose that the probability of breaking your

leg at a ski lodge is 0.20

1. What is the probability that exactly 2 out of 5 people

will break their leg?

2. What is the probability that more than 2 people out of 5

Example II - Binomial Distribution

1. What is the probability of obtaining exactly 4

heads in 6 flips of a fair coin?

2. What is the probability of obtaining 4 or more

Example III - Binomial Distribution

• Express House Delivery Service guarantees a refund of

all charges if delivery does not arrive by the specified time. It is known that 2% of packages do arrive by the

specified time. Suppose a corporation mails 10 packages on a certain day.

• A) Find the probability that exactly one of these ten

packages will not arrive by the specified time.

• B) Find the probability that at most one of these ten

Expectation & Variance – Binomial

Distribution

• If the random variable X is binomially distributed

then:

• E(X) = np

• Var(X) = npq or np(1-p)

Example – Binomial Expectation &

Variance

• Suppose experience has shown that 30% of the

television sets sold on hire purchase at a certain store are eventually re-possessed. If 10 TV sets are sold during a week, determine:

NORMAL DISTRIBUTION

(CONTINUOUS PROBABILITY

DISTRIBUTION)

■ What is the Normal Distribution (ND)?

■ The Standard Normal Distribution & Its Probabilities ■ How to Calculate General Probabilities for the ND ■ The Normal Approximation of the Binomial

Distribution

The Normal Distribution

• The normal distribution is a continuous

random variable

• It is symmetric and bell-shaped

• The mean = median = mode

• Shape of curve depends on population

The Normal Distribution

• Center of distribution is μ

- the mean

• Spread is determined by

σ - the standard

deviation

• 50% of the scores lie

above the mean and 50% lie below the mean

• Total area under the

General Normal Distributions

•

The mean and standard deviation affect the

shape of the normal distribution

Standard Normal

Smaller Standard Deviation

The Standard Normal Distribution

•

This is a special case of the normal distribution

•

Mean = μ = 0

•

Standard Deviation = σ =1

•

The Normal Distribution Table calculates these

Distribution

• The units for the standard normal distribution

curve are denoted by z and are called z values

• The z-value gives the distance between the mean

and the point represented by z in terms of the standard deviation

The Standard Normal Tables

•

The tables give you

the area to the left

of any positive value

•

The area under the

curve is equal to 1

•

The area to the right

is therefore 1 minus

Normal Tables cont’d

Normal Tables cont’d

Steps to Calculate Normal

Probabilities

⚫

Step 1

: Sketch the normal distribution and indicate

the mean (the middle) of the random variable X

⚫

Step 2

: Shade the area you want to find

⚫

Step 3

: Find the corresponding area to the right of the

mean (if needed)

Example I – Normal Distribution

1. What is the probability of X being less than 1.5

2. What is the probability of X being greater than

1.5?

3. What is the probability of X being less than

-0.67?

4. What is the probability of X being greater than

Example II – Normal Distribution

5. What is the proportion that falls between 0.5 and 1.5?

6. What is the proportion that falls between -0.5 and 1.5?

Standardization

• To calculate probabilities from a normal

distribution with μ ≠ 0 and σ ≠ 1 (ie. not standard

normal) we have to standardize the distribution

• Once we know the mean and standard deviation

this can be done

• The formula is:

• After the standardization process is complete (ie.

Example - Standardisation

•

Assume that the length of time, X, between

charges of a cellular phone is normally

distributed with a

• mean of 10 hours and a

• standard deviation of 1.5 hours

Find the probability that the cellular phone

will last for :

a. Less than 13 hours between charges b. More than 12.5 hours between charges

c. Between 8 and 12 hours between charges d. Between 6 and 9 hours between charges

Calculating % aspects

⚫ Sometimes the probability (area under the curve) is given for us to determine the value

⚫ Step 1: Draw the diagram and shade the relevant area

⚫ Step 2: Determine the standardised value

⚫

To qualify for the police academy, candidates must

Example – Normal Distribution % II

•

It is known that the life of a calculator

manufactured by Texas Instruments has a

normal distribution with a mean of 54 months

and a standard deviation of 8 months. What

should the warranty period be to replace a

using the Normal Distribution I

• When n is large (n ≥ 20), a normal probability

distribution may be used to provide a good

approximation to the probability distribution of a binomial random variable

Approximating the Binomial Distribution

using the Normal Distribution II

⚫The Continuity Correction Factor (CCF)

■ This is the value which is added when a discrete

distribution (eg. The binomial distribution) is being

approximated by the normal distribution (a continuous distribution)

■ Ie. we are correcting the discrete distribution so that it

can be approximated by the continuous one

■ The CCF value used is 0.5

■ Subtract 0.5 from lower limit and add 0.5 to upper

Approximate the Binomial Distribution I

⚫ Step 1: Calculate the mean and variance of the binomial distribution in question

Approximate the Binomial Distribution II

•

Step 3: Calculate the probability to be

approximated using the formula:

•

Step 4: Sketch the approximating normal

distribution and shade the area corresponding

to the probability of the event of interest

•

Step 5: Find the probability using the ‘Steps

to Calculate Normal Probabilities’ slide

The mean of the binomial distribution

to the Binomial Distribution

•

In a recent survey conducted for Money

magazine, 80% of the women surveyed said

that they are more knowledgeable about

investing now than they were five years ago

(Money, June 2002). Suppose this result is

true for the current population of all women.

What is the probability that in a random

sample of 100 women, 72 – 76 will say that

Example II – Normal Approximation

to the Binomial Distribution

•

According to the 2001 Youth Risk Behaviour

Surveillance by the Centres for Disease Control and

Prevention, 39% of the 10

-graders surveyed said

that they watch three or more hours of television on a

typical school day. Assume that this percentage is

true for the current population of all 10

-graders.

What is the probability that 86 or more of the

Binomial vs. Normal Distributions

Binomial Normal

Variable Type Discrete (assumes only discrete

values)

Continuous (assumes all values within a given interval)

Mean np μ

Variance npq σ2

Symmetry

■ Only symmetrical when p = 0.5

■ If p = 0.5 the distribution is

skewed but tends to symmetry as n increases