Random Variables and its Probability Distributions

What is a Random Variable?

A Random Variable or Stochastic Variable or Random quantity in the field of probability and statistics is a variable quantity, whose possible values depend on a set of random outcome events in random manner.

The outcome space is defined in terms of set theory or a set.

A random variable is a function associating a real number with each outcome of a sample space S of a random experiment and it is denoted by X.

Domain of a random variable X is a sample space S and codomain or range of a random variable X is the set of real values taken by X, Thus, symbolically X: S-> R.
 

Types of Random Variable

There are two types of Random Variable:

• Discrete Random Variable

• Continuous Random Variable

Discrete Random Variable

A random variable X which can assume countable number of isolated values (integers) is called discrete random variable. The meaning of the word ‘discrete’ is separate and individual

Examples:

• Number of kids in a family

• Number of attempts to hit a target

• Number of heads in tossing a coin thrice

• Number of stars in the sky
   

Continuous Random Variable

A random variable X which can assume all real values within a given interval is known as continuous random variable. Thus, the possible values of a continuous random variable are uncountably infinite.

Examples:

• Price of a commodity

• Age of a person

• Height of a person in cm

• Life of an electric component (in hours)
   

Examples to identify random variables as either discrete or continuous in each of the following situations:

Q1.  A  page in a book can have at most 300 words

X= Number of misprints on a page

Solution

X = Number of misprints on a page

Since a page in a book has at most 300 words, X takes the finite values

Therefore, random variable X is discrete

Range = {0, 1, 2….299, 300}
 

Q2. A gymnast goes to the gymnasium regularly

X = Reduction of his weight in a month

Solution

X = Reduction of weight in a month

X takes uncountable infinite values

Therefore, random variable X is continuous.
 

Probability Mass Function

If X is the discrete random variable taking values x₁, x₂, x₃…x_n and we find the values of P(X = x₁), P(X = x₂), P(X = x₃)…P(X = x_n), then we obtain the functions P(x) = P(X = x) for x = x₁, x₂, x₃…x_n.

This function is called Probability Mass Function (p.m.f) of the random variable X.

Example

If two coins are tossed and X=number of heads, then the sample space of tossing two coins is {HH, HT, TH, TT}

Therefore, P(X = 0) = P (0) =1/4, P(X = 1) = P(1) = 2/4 = 1/2

P(X = 2) = P (2) = 1/4

Hence, the probability mass function is given by

P(x) = 1/4, x = 0

=1/2, x = 1

= ¼, x = 2

For the sample space S = {0, 1, 2} for X.

Note:

If X is any random variable taking values from sample space S and P(x) is the probability mass function of X such that x ∈ S, then

• 0 ≤ P(x) ≤  1 for all x ∈ S

• ∑ P(x) = 1, that is, the total of all values of P(x) for x ∈ S is always unity.

We can verify this in the example given above

∑ P(x) = P(0) + P(1) + P(2) = 1/4 + 1/2 + ¼ =1
 

What is Probability Distribution?

If X is a discrete random variable taking values x₁, x₂, x₃….x_n with corresponding probabilities p₁, p₂, p₃…p_n, then the set of ordered pairs (x_i, p_i), i = 1, 2, 3….n is called a Probability Distribution of the random variable X.

Example

If the two coins are tossed and X is the number of heads, then the probability distribution of the random variable X can be given as below:

Note that the sum of all probabilities

 = ¼ + 2/4 + ¼ = 4/4 = 1

In general case, the probability distribution of discrete random variable X can be given as:

Where ∑ p_i = p₁ + p₂ + p₃ +….+ p_n = 1
 

Cumulative Distribution Function (C.D.F) of Discrete Random Variable

Let X be a discrete random variable and its probability distribution is as follows:

The Cumulative Distribution Function (c.d.f) of X is denoted by F(x) and it is defined as

F(x) = P[X ≤ x], x ∈ R

Notes:

• Domain of c.d.f is R

Codomain of c.d.f is [0, 1]

• F(x_i) = P[X ≤ x_i] = p₁ + p₂ + p₃ +….p_i, i = 1, 2, 3,…n

• c.d.f of a discrete random variable X is also represented in tabular form as follows:

• C.D.F. is often called Distribution Function (D. F)
   

Problems

Q1. Three balanced coins are tossed simultaneously. If X denotes the number of heads, find probability distribution of X.

Sol.  When three balanced coins are tossed then the sample space is

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

X denotes the number of heads.

X can take the values 0, 1, 2, 3

P[X = 0] = P (0) = 1/8
P [X = 1] = P (1) = 3/8

P[X = 2] = P (2) = 3/8
P [X = 3] = P (3) = 1/8

Q2. Given below is the probability distribution of X:

• Find the value of k

• P (X ≥  2), P(X < 3 ), P(X≤1)

• Obtain the c.d.f of X

Solution:

• Since P(X) is the probability distribution of X,

∑ P[X = x] =1

P[X = 0] + P[X = 1] + P[X = 2] + P[X = 3] + P[X = 4] = 1

k + 2k + 4k + 2k + k =1

10k = 1

k = 1/10

• P (X ≥ 2) = P[X = 2] + P[X = 3] + P[X = 4]

= 4k + 2k + k = 7k

= 7 (1/10) = 7/10

P (X<3) = P[X=0] +P [X=1] + P[X=2]

= k + 2k+ 4k = 7k

= 7 (1/ 10) = 7/10

P (X≤ 1) = P[X=0] + P [X=1]

= k +2k = 3k

= 3 (1/10) = 3/10

• By definition of C.D.F,

F(x) = P (X ≤ x)

F(0) = P (X ≤ 0) = P(0) = k = 1/10

F(1) = P(X ≤ 1) = P(0) + P(1) = k + 2k = 3k = 3/10

F(2) = P(X ≤ 2) = P(0) + P(1) + P(2) = k + 2k + 4k = 7k = 7/10

F(3) = P(X ≤ 3) = P(0) + P(1) + P(2) + P(3) = k + 2k + 4k + 2k = 9/10

F(4) = P(X ≤ 4) = P(0) + P(1) + P(2) + P(3) + P(4) = k + 2k + 4k + 2k + k = 10/10 = 1

Hence, the c.d.f of X is as follows:

Probability Density Function (P. D. F) and Distribution Function (D. F) of a Continuous Random Variable
 

Probability Density Function

  A real valued function f(x) is called a Probability Density Function (P. D. F) of a continuous random variable X, if it satisfies the following:

• F(x) ≥ 0 for all x ∈ R

Notes:

• If X takes the values in the interval (a,b) then the function f(x) is such that

  • F(x) ≥ 0 for a < x <  b, and

  • ∫^b_a    f(x) dx =1

• The geometrical representation is as below:

The area under the curve y= f(x) bounded by the X-axis and the coordinates x = a and x = b is 1, because it represents the total probability P(a< X <b) which is equal to 1.

Also, P (p < X < q) is area under the curve y = f(x) bounded by the X-axis and the coordinates x = p and x = q which is shaded in the figure.

The graph of the P. D. F of R. V. X is called the Probability Curve or Probability Density Curve.
 

Distribution Function

Assume X as a continuous random variable with probability density function f(x), then the cumulative distribution function F(x)of X is defined for every real number x_i, as F(x_i) =P [X ≤ x_i] = ∫ f(x) dx

Remarks:

• F(x_i) is the area under the curve y= f(x) to the left of x as shown in the figure right:

• F(x) increases smoothly as x increases

• If range of X is (a,b) then F(x) = 0, for x< a and F(x) = 1, for x≥b

• P [X >x]=1 - P [X ≤ x_i]= 1- F(x)
   

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Random Variables and its Probability Distributions