Random Variables:

A random variable is the value of the variable which represents the outcome of a statistical experiment within sample space (range of values). It is usually represented by X. The two types of random variables are discrete random variable and continuous random variable.

Discrete random variable is a variable which can take countable number of distinct values.

Example:

while tossing two coins, let us consider the random variable(X) to be number of heads observed.

Here, the possible outcomes are {HH, HT, TH, TT} which means the number of heads possible in a single outcome may be 2 heads or 1 head or no heads at all. Hence, the possible values taken by the random variable are 0, 1, and 2 (discrete).

Continuous random variable is a variable which can take infinite number of values in an interval.  It is usually represented by the area covered under the curve. Examples are height and weight of the subjects, maximum and minimum temperatures of a particular place.

Let Y be the random variable for the average height of a random group consisting of 25 people,  the resulting outcome is a continuous figure since height may be 5 ft or 5.15 ft or 5.002 ft. Clearly, there is an infinite number of continuous possible values for height.

Properties of Random variable

• Let X and Y be two random variables and the constant C. Then CX, X+Y, X-Y are also random variables. Any arithmetic operation with random number results in another random variable.
• If X is a random variable, then  are also said to be random variables.

A random variable’s behavior is defined by a probability distribution which is the likelihood that any of the possible values would occur.

Let Z be the random variable which is the number on the top face of a die when it is rolled once. The possible numbers for Z are 1, 2, 3, 4, 5, and 6. P(1)=P(2)=P(3)=P(4)=P(5)=P(6) = 1/6 as they are all equally likely to be the value of Z. Note that the sum of all probabilities is 1.

Suppose in a probability distribution where the outcomes of a random event are not equally likely to happen we can still find the probabilities of random variable. Let Y be random variable which is the number of heads we get from tossing two coins, then Y could be 0, 1, or 2. The two coins will flip in 4 possible different ways – TT(no heads), HT(one head), TH(one head), HH(2 heads).

So, P(Y=0) = ¼ as we have only one chance of getting no heads (TT).

Similarly, P(Y=2) = 1/4 (HH).

But,  P(Y=1) = 2/4 = ½.(HT,TH)