18thJuly_RandomVariables_ProbDistributions

7/29/2019 18thJuly_RandomVariables_ProbDistributions

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Random Variables and

Probability Distributions(partial)

Ravindra S. Gokhale

IIM Indore

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Random Variables

A variable that associates a number with the outcome of a

randomexperimentis a random variable

Denoted by an uppercase letter such asX,Y, etc.

A random variable can take only numeric values.

Toss of a coin is NOT a random variable.

[It is an experiment that yields random results]

However, number ofheads from toss of a coin is a random

variable

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Random Variables

Examples:

If two dice are thrown, the sum of the faces is a random variable, as in:

X = 3, X = 11, etc.

If two coins are tossed, then the number of heads is a random variable,

as in: X = 0, X = 2, etc.

In a speed (rpm) measurement: X = 457, X = 1209, etc.

In a dimension measurement with the help of a caliper: X = 23.46, X =

48.97, etc.

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Random Variables (cont)

A random variable with a finite or countable infinite range

Examples:

Number of scratches on a car surface

Proportion of defective parts among 1000 tested

Number of people arriving at a bank in a given time interval

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Discrete Random Variables


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A random variable with an interval (either finite or infinite) of real

numbers for its range

Examples:

Length dimension (likesurface area of a table)

Time dimension (liketime between failure for a machine)

Temperature dimension (liketemperature inside a room)

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Continuous Random Variables


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Population in a particular state of India.

Total weight of consignments handled by a courier company in a

day.

Time to complete an exam.

Number of participants in an exit poll.

Total number of goals scored in a football game.

Life of a particular medicine.

Height of the Ocean's tide at a given location.

Amount of rain on a particular day.

Number of train derailments in a year.

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Discrete or Continuous?


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The manner in which random variables are expressed sometimes

depends on the problem at hand

Sometimes a random variable is discrete in nature, but it is

treated continuous

This is because the range of values it can take is too large

Example: Marks of a student in a 100 marks paper

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Expression of Random Variables


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Sometimes a random variable is continuous in nature, but it is

treated discrete

This is because the exact value (to the smallest level) is not required

Example: Age of a person may be expressed as a discrete random

variable forming different categories: 0-21, 21-35, 35-50, 50-65, 65+

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Expression of Random Variables


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Probability Distributions

Probability distribution of a random variable X is a formula, table,

or graph that gives all possible values of X and corresponding

probabilities P(X = x) for all x's in the domain of X.

Example: Probability distribution of roll of a dice:

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x 1 2 3 4 5 6

P(X = x) 1/6 1/6 1/6 1/6 1/6 1/6


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Probability Distributions

Standard probability models (probability distributions) are

available in the literature and have been studied in detail.

These models can mimic many real life scenarios very well and

have mathematically tractable representation.

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Discrete Random Variables

Examples:

If two coins are tossed and we are interested in the event number of

heads obtained, then:

P(X = 0) = 0.25 P(X = 1) = 0.50

P(X = 2) = 0.25 P(X = 3) = 0

P(X > 1) = 1 [P(X = 0) + P(X = 1)] = 1 (0.25 + 0.50) = 0.25

In a lot that contains 10% defective pieces, if we are interested in the

number of defective pieces in a sample of 5then:

P(X = 0) = 0.590 P(X = 1) = 0.328

P(X = 2) = 0.073 P(X = 3) = 0.008

P(X = 4) 0.001 P(X = 5) 0.000

P(X


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Discrete Random Variables (cont)

Terminologies associated with discrete random variables

Probability mass function (pmf) denoted by f(x)

Cumulative distribution function (cdf) denoted by F(x)

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x

f(x)

Probability mass functionof a fair dice

f(xi) = P(X = xi)

1 2 43 650

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2/6

1

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x

F(x)

Cumulative distribution function ofa fair dice

F(x) = P(X


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Mean of a discrete random variables

Mean is the expected value of the random variable denoted by

or E(X)

It is the measure of thecenterof the probability distribution

Formula:

If we make infinite number of draws from the distribution of a

random variable and calculate the average of the data then the

average is the expected value (or mean) of the random variable.

Note: The expected value should not be confused with most

likely value.13

= E(X) = x f(x)x

Mean and Variance of a Discrete Random Variables


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A simple example:

You can insure a Rs.500,000 jewellery against theft for its total

value by annual premium of Rs. R. If the probability of theft in a

given year is estimated to be 0.01, what premium should the

insurance company charge if it wants an annual expected gain

equal to Rs. 10,000?

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Variance of a discrete random variables

Denoted by 2 or V(X)

It is the measure of the dispersion or variability in the probability

distribution

Formula:

The standard deviation () of X is the (positive) square root of the

variance

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2= V(X) = (x )2f(x) = [ x2 f(x)] 2xx


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Continuous Random Variables

Examples:

When a machine breaks down, it is serviced. It runs for some time until

it again breaks down. We are interested in the event time (in hours)

between successive breakdowns, then:

P(X < 10) = ? P(X > 250) = ?

P(50 < X < 150) = ?

A finance executive wants to predict the various financial ratios (say X,

Y, etc.) of different organizations, based on past data. For a particular

organization, he may be interested in:

P(X > 0.75) = ? P(Y < 0.6) = ?

P(0.35 < X < 0.50) = ?

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Terminologies associated with continuous random variables

Probability density function (pdf) denoted by f(x)

Cumulative distribution function (cdf) denoted by F(x)

Continuous Random Variables (cont)

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pdf

Resembles a histogram

Used to calculate an area that

represents the probability that X

takes the values between [a, b]

P(a


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Continuous Random Variables (cont)

Mean of a continuous random variables

Defined similarly to that of a discrete random variable

Denoted by or E(X)

Formula:

Variance of a continuous random variables

Defined similarly to that of a discrete random variable

Denoted by 2 or V(X)

Formula:

The standard deviation () of X is the square root of the variance

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Mean and Variance of a Continuous Random Variables

dxf(x)][xE(X)

2222 -dx}f(x)][x{dx}f(x)]-[x{V(X)


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Problems

The probability density function of the length of a metal rod is f(x)

= 2 for 2.3 < x < 2.8 meters

If the specifications of this process are 2.25 to 2.75 meters, what

proportion of the rods fail to meet the specifications?

Determine the mean and the variance of the length of the metal rod

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pdf, mean and variance of Continuous Random Variables


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Bernoulli Trial

A basic building block for all the discrete probability distributions

A trial has only two possible outcomes

Usually termed as:successandfailure

Examples:

Did tossing of a coin lead to a head (success) or not?

Did the student pass the exam (success) or not?

Did India lose the match (success) or not?

Was the part defective (success) or not?

Probability of success is denoted byp

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Bernoulli Trial

Mean of a Bernoulli Trial = p

Variance of a Bernoulli Trial = p (1 p)

Derive

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