Random Variable and Probability Distribution

If is often very important to allocate a numerical value to an outcome of a random experiment. For example consider an experiment of tossing a coin twice and note the number of heads (x) obtained.

Outcome : HH HT TH TT

No. of heads (x) : 2 1 1 0

x is called a random variable, which can assume the values 0, 1 and 2. Thus random variable is a function that associates a real number to each element in the sample space.

Random variable (r.v)

Let S be a sample space associated with a given random experiment.

A real valued function X which assigns to each wi Î S, a unique real number.

Note:

There can be several r.v's associated with an experiment.

A random variable which can assume only a finite number of values or countably infinite values is called a discrete random variable.

e.g., Consider a random experiment of tossing three coins simultaneously. Let X denote the number of heads then X is a random variable which can take values 0, 1, 2, 3.

Continuous random variable

A random variable which can assume all possible values between certain limits is called a continuous random variable.

Discrete Probability Distribution

A discrete random variable assumes each of its values with a certain probability,

Let X be a discrete random variable which takes values x₁, x₂, x₃,…x_n where p_i = P{X = x_i}

Then

X : x₁ x₂ x₃ .. x_n

P(X): p₁ p₂ p_{3 .....} p_n

is called the probability distribution of x.

Note 1:

In the probability distribution of x

Note 2:

P{X = x} is called probability mass function.

Note 3:

Although the probability distribution of a continuous random variable cannot be presented in tabular form, it can have a formula in the form of a function represented by f(x) usually called the probability density function.

Probability distribution of a continuous random variable

Let X be continuous random variable which can assume values in the interval [a,b].

A function f(x) on [a,b] is called the probability density function if

Mean and Variance of a Discrete Random Variable

Let X be a discrete random variable which can assume values x₁, x₂, x₃,…x_n with probabilities p₁, p₂, p₃ ….. p_n respectively then

(a) Mean of X or expectation of X denoted by E(X) or m is given by

(b) Variance of X denoted by s² is given by

Note :

Example:

Two cards are drawn successively without replacement, from a well shuffled deck of cards. Find the mean and standard deviation of the random variable X, where X is the number of aces.

Suggested answer:

X is the number of aces drawn while drawing two cards from a pack of cards.

The total ways of drawing two cards ⁵²C₂. Out of 52 cards these are 4 aces. The numbers of ways of not drawing an Ace =⁴⁸C₂. The number of ways of drawing an ace is ⁴C₁ x ⁴⁸C₁ and two aces ⁴C₂.

Therefore the r.v. X can take the values 0, 1, 2.

= 0.1629 - 0.0236

= 0.13925

Let X be a continuous random variable which can assume values in (a, b) and f(x) be the probability density of x then

(a) Mean of X or Expectation of X is given by

(b) Variance of x is given by