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 wi
Î S, a unique real number, X(wi)
= xi 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,…xn where pi = P{X = xi}
Then X : x1 x2 x3 …xn
P(X) : p1 p2 p3 … pn 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.
