Random Variables and Probability Distributions

Ask a Question, Get an Answer!

Type your question here

Hundreds of tutors are online and ready to help you right now!

It 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  w_i Î S, a unique real number, X(w_i) = x_i is called a random variable.

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 obtained. Then, X is a r.v 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 p₃ … p_n is called the probability distribution of x,

If in the probability distribution of x,

Note 1 :

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

Note 2:

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