Introduction To Probability Distribution Lecture 21

Introduction

To

Probability Distribution

Lecture 21

Random Variable

In probability, the random variable is commonly used to quantify the results of random events. Random variables, often known as chance variables, are numerical quantities whose values are determined by the results of a random experiment. The random variable is denoted by capital letters like X, Y, etc., and specific values are denoted by small alphabets like x, y, etc.

Suppose we are interested in the number of heads when two coins are tossed once.

Then the sample space is given by:

S: HH    HT    TH   TT

X (Number of Heads):   2    1     1     0

Thus, X = 0, 1, 2 is a random variable.

Types of Random variables

1.      Discrete Random Variable

2.      Continuous Random Variable

Discrete Random Variable

A random variable that can take on specific finite or countable infinite values.

Examples:

i.                    The number of heads (or tails) in coin experiment.

ii.                  The sum of dots in a rolling dice experiment.

iii.                The number of boys (or girls) in three child family, etc.

Continuous Random Variable

In contrast to discrete random variables, continuous random variables can have an unaccountably infinite number of possible values inside a certain interval (a, b). where an is less than b.

Examples:

i.                    The temperature of a locality between 6.00 am and 9.00 pm.

ii.                  The amount of rain recorded in the month of July.

iii.                The weight of new born babies, etc.

Probability Distribution

A probability distribution is created when the values of a random variable "X" and the probabilities associated with them are presented in tabular form. Let X1, X2, …, Xn be the values of a random variable and f(x1), f(x2), …, f(xn) such that the sum of all probabilities is one.

That’s

X	X1	X2	….	Xn
f(x)	f(x1)	f(x2)	….	f(xn)	1

The discrete probability distribution is graphically represented by a probability histogram.

Properties:

i.f(x)≥  0  ∀ x

ii. ∑f(x) = 1

Probability Function

If an equation is used to represent the values of a random variable, their respective probabilities are called a probability function.

Mathematically, it can be written as:

f(x) = P (X = x) X = 0, 1,...

If a random variable is discrete, then it is called the probability mass function, and if a random variable is continuous, then it is called the probability density function.

Properties:

i.f(x)≥  0  ∀ x

ii. ∑f(x) = 1

Example 6.1: Find the probability distribution and probability function for the number of heads when two coins are tossed.

Solution: The number of sample points in S = 4.

Let X be used to represent the number of heads.

The probability distribution is given by:

X	0	1	2
f(x)	1 / 4	2 / 4	1 / 4	1

The number of ways in which x number of heads appears when 2 coins are tossed is given by:

The sample space is given by:

The probability function is given by:

Example 6.2: Find the probability distribution and probability function for the number of heads when three coins are tossed.

Solution: The number of sample points in S = 8.

S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

n(S) = 8

Let X be used to represent the number of heads.

The probability distribution is given by:

X	0	1	2	3
f(x)	1 / 8	3 / 8	3 / 8	1 / 8	1

Now to obtain the probability function as:

The number of ways in which x number of heads appears when 3 coins are tossed is given by:

The sample space is given by:

Example 6.3: Find the probability distribution and probability for the sum of dots when two dice are rolled.

Solution: The number of sample points in S = 36.

Let X be used to represent the sum of dots.

Tabulation of sample space of the sum of dots is given by:

The probability distribution of the sum of dots is given by:

X	2	3	4	5	6	7	8	9	10	11	12
f(x)	1 / 36	2 / 36	3 / 36	4 / 36	5 / 36	6 / 36	5 / 36	4 / 36	3 / 36	2 / 36	1 / 36	1

Example 6.4: A bag contains 4 red and 3 white balls. Two balls are selected at random. Find the probability distribution and probability function of the number of red balls.

Solution: The sample space will be:

Let X represent the number of red balls

The probability distribution of the number of red balls (X) is given by:

X	0	1	2
f(x)	3 / 21	12 / 21	6 / 21	1

Now to obtain the probability function as:

The number of ways in which x number of red balls and selected minus red ball (i.e., 2 – x) is the number of non-red (white balls) is selected:

The sample space is:

The probability distribution for the number of red balls is given by:

 

Example 6.5: Two cards are selected from a deck of 52 playing cards. Let X represent the number of kings. Find the probability distribution of the number of red balls.

Solution: The sample space is given by:

Let X represent the number of kings.

The probability distribution of the number of kings is given by

• Read More: Mathematical Expectation