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Random Variables and its Probability Distributions

 

Table of Content


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.

Random VariableThe 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.

Definitions Random VariablesExamples:

  • 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…xn and we find the values of P(X = x1), P(X = x2), P(X = x3)…P(X = xn), then we obtain the functions P(x) = P(X = x) for x = x1, x2, x3…xn.

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….xn with corresponding probabilities p1, p2, p3…pn, then the set of ordered pairs (xi, pi), 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:

X 0 1 2
P(X) ¼ 4-Feb 4-Jan

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:

X X1 X2 X3 …. Xn
P(X) P1 P2 P3 …. Pn

Where ∑ p= p+ p+ p+….+ p= 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:

X = xi X1 X2 X3 Xn Total
P= P[X = xi] P1 P2 P3 Pn 1

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(xi) = P[X ≤ xi] = p+ p+ p+….pi, i = 1, 2, 3,…n

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

X=xi F(xi)
X1 P1
X2 P1+P2
X3 P1+P2+P3
……. …….
Xn P1+P2+P3+…..+Pn
  • 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:

X 0 1 2 3 4
P [X= x] k 2k 4k 2k k
  • 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:

xi 0 1 2 3 4
F(xi) 10-Jan 10-Mar 10-Jul 10-Sep 1

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

    • ba    f(x) dx =1

  • The geometrical representation is as below:

The Geometrical Representation

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 xi, as F(xi) =P [X ≤ xi] = ∫ f(x) dx

Remarks:

  • Distribution FunctionF(xi) 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 ≤ xi]= 1- F(x)
     

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Random Variables and its Probability Distributions

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