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CHAPTER 2
Random Variable
A random variable is essentially a random number
...
If
the random variable is given by a capital letter, the corresponding small letter denotes the
various values that the random variable can take on
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2
...

Events are often expressed as random variables in statistic, and we have to calculate
probabilities for these events (probability of the random variable)
...
The probability mass function (or frequency
function) of X is defined as the function
such that:
or

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The
height of each bar equals the probability of the corresponding value of the random variable
...

Sample space:
i)

Let the random variable

denote the number of heads
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The probability mass function is then:

The probability mass function (frequency function) is represented by a bar chart in
Rice, p
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ii)

Let the random variable Y denote the number of tails
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The probability mass function is then:

iii)

Let the random variable Z denote the number of heads minus the number of tails
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Example 2
Two socks are selected at random from a drawer containing five brown socks and three
green socks
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The values (mass points) that the random variable can take on is


...


and
Thus, the given function can serve as the frequency function of the random variable with
values {1,2,3,4,5}
...

Suppose X is a discreet random variable
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3

, then

Example 4 (continuation of Example 1)
Find

Cumulative distribution function (cdf): The cumulative distribution function of a random
variable X is defined as:

...


Observe that this distribution is defined not only for the values taken on by the given
random variables, but for all real numbers
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The cdf is shown graphically in Rice, p
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4

and


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The distribution function can also be used to obtain the mass function (frequency function)
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Next we look at a number of important discrete random variables
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1
...
1
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Let denotes the number of
successes obtained from these n Bernoulli trails
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Properties of a binomial random variable:
i)

The experiment consists of

independent and identical trails
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iii)

The probability of a success for a single trail is

and the probability of a failure is


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Is
variable?

a binomial random

Binomial probability mass function: Let denote the probability of success in a Bernoulli
trial
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The
probability of this realization is (from independence of Bernoulli trials and the multiplicative
rule):

...
Therefore

7

ways of choosing

success

Properties of the binomial probability mass function:
i)
ii)
Example 11
i)

Find the probability of getting 5 heads in 12 flips of a coin
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iii)

Find the probability of getting 2 or more heads in 12 flips of a coin
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8 that any one of them will recover
from the disease
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See Examples A and B (Rice, p
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8

Note: Let
Then

be independent Bernoulli random variables with
is a binomial random variable
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2
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3 Geometric Random Variable
Assume repeated independent Bernoulli trials, where is the probability of success at each
trail
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is called a
geometric random variable
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02
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Then is a geometric random variable with
, and

See Example A (Rice, p
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9

Negative Binomial random variable
The negative binomial distribution is a generalization of the geometric distribution
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If is the number of trials until r successes are obtained then is
a negative binomial random variable
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See Example B (Rice, p
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Example 15
The probability is 0
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What is the
probability that the tenth child exposed to the disease will be the third to catch it?

Example 16
If a coin is tossed, what is the probability that with the 4 th toss we have observed 2 heads?

2
...
4 Hypergeometric random variable
10

Suppose a container holds balls, where of them are black and
are white
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Let be the number of black balls drawn in this
way
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Similar to the Binomial distribution but it only applies to sampling without replacement
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Find the probability to obtain

successes in

successes and there are

ways to choose

Therefore, there are
are

ways to choose

trails: There are

ways to choose
of the

Therefore,

of the
successes and

ways to choose

of the

failures
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...
Let
number of black balls drawn when taking 3 balls without replacement
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Example 18
As part of an air-pollution survey, an inspector decides to examine the exhaust of six of the

See Example A (Rice, p
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11

2
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5 Poisson random variable
-

Poisson probability mass function:
parameter

with


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Assume that the number of accidents in a week follow a
Poisson distribution with

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The average number of accident per week is
in a 1 week period
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Let X be the number of accidents

Calculate the probability that at most three accidents occur during a two week
period
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Example 20
A hospital switchboard receives emergency calls at a rate of 2 every 30 minutes
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...
46)
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The binomial frequency function is

Setting

the frequency function becomes

As
, therefore

Proof: Let


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Note: Suppose is binomial with parameters
becomes large and small (with
):

The Poisson approximation works well if

and , then it follows from above that as

and


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The screws are
packed into boxes which can hold 50 screws each
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Let X be the number of screws that are defective, then X is a binomial random variable with
and

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Then, by making use of the Poisson approximation:

Check this answer by calculating the exact binomial probability
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44)
...
2

Continuous random variables

Continuous random variables take on values on a continuous scale, such as heights, weights
and length of life of a particular product
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If you try to assign a positive probability to each of these uncountable values, the
probabilities will no longer sum to 1, as with discreet random variables
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For a continuous random variable the role of the frequency function is taken by the density
function (pdf)

...
04

0
...
035

0
...
03

0
...
025

0
...
02

0
...
015

0
...
01

0
...
005

0
...

Consider the probability that x equals some particular value, say
...


Therefore

15

Cumulative distribution function: The cdf of a continuous random variable is defined just as
it was in the discrete case:

There exists a one-to-one relationship between distribution functions and density funtions:
i)

(by definition)

ii)
There is also a relationship between distribution functions and probabilities:

Example 22
If

has the probability density

Find

and


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ii)
Example 23
Find the cdf of the random variable X in the previous example, and use it to re-evaluate

...

a)

Find

b)

Find cdf and recalculate (a)

Example 25
Show that

represents a pdf and calculate

i)
ii)
Example 26
The density function of a random variable Y is given by

a) Find

b) Find

c) Find the cdf and recalculate a and b
17


...


Uniform density function (i): A uniform random variable
has a density function given by

on an interval [0, 1] (

)

on an interval [a, b] (

)

f(x)
1
0

1

Example 27

Uniform density function (Ii): A uniform random variable
has a density function given by

18

1/(b-a)
a

b

Example B (Rice p
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i)

What is the probability that a random patient dies before 8am on the day of their
death?

ii)

What is the probability that a random patient dies during visiting hours, if visiting
hours are 10am to noon?

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iii)

What is the probability that a random patient dies during visiting hours, if visiting
hours are 10am to noon and 6pm to 9pm?

iv)

What is the probability that a random patient dies at exactly midnight?

pth Quantiles: The pth quantile of a df

is defined as the number

such that

therefore

is called the first quartile (lower quartile)
is called the second quartile (median)
is called the third quartile (upper quartile)
See Example C (Rice, p
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Exponential density function: An exponential random variable
density function

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with parameter

has a

2
...
5

1

0
...
2

0
...
6

0
...
2

1
...
6

1
...
9
0
...
7

F(x)

0
...
5
0
...
3
0
...
1
0

0

5

10

15
x

20

25

30

It is easy to obtain the pth quantile of :

Then by definition

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Thus

The memoryless property of the exponential density:
The exponential distribution is used to model lifetimes or waiting times
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Suppose
also that the component has already lasted for a certain period of time , and that now we
want to determine the probability that the component will last for at least another time
units, that is, we want to calculate

...


This is called the memoryless property of the exponential distribution
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A person enters a room where a light bulb has

already been burning for three hours
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What is the probability that the person will get their work done before the light bulb
expires?

22

Example 30
A petrol station in a remote town gets very few customers
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Suppose a customer has just left the petrol station
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i)

Give the density function of the inter-arrival time
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is the shape parameter and is the scale parameter
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54)
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is called the standard deviation and it

The cdf cannot be evaluated in close form (the integral over the pdf cannot be evaluated
explicitly)
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Note: If

and


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13, Rice, p
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Beta distribution
Density function of the Beta random variable: A continuous random variable X is beta
distributed with parameters and if it has the following density function:

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The beta density is useful for modelling random variables that are restricted to the interval
[0,1]
...
3

is the uniform distribution
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We might be interested in
finding the density function of another random variable Y which is a function of X:

...



...

Proposition A, Rice p
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Proof:
Differentiate on both sides to y to get the density function

Therefore
Result:
Suppose

and let

Then


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Suppose now that we want to calculate

is

25

:


...


We read the probabilities of the standard normal distribution from the standard normal
table
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60 p
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62: Let X be a continuous random variable with density function
and let
where is a differentiable, monotonic function
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62)
Let U be uniform on [0, 1],


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Determine the density

Example 31
Suppose X has density function


...


Differentiate of both sides to Y:

Alternative: (Using Proposition B)

Example 32
The density function of X is:


...




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