# Random variable and its probability distribution

A random variable is defined as a function that assigns a real number to each outcome in a sample space in the case of a random experiment. It is denoted by an uppercase letter, X while the measured value of the random variable is denoted by a lowercase letter, x. A discrete random variable lies on a countable or finite range while a continuous random variable lies with in an interval of real numbers of its range that is finite or infinite. If n elements are selected from a population under consideration such that every set of n element in the population has an equal probability of being selected, then we can say that n elements are said to be a random sample.

The probability distribution for a random variable explains probabilities distributed over the values of random variable.The probability distribution is defined by a probability function, denoted by P(x), which provides the probability for each value of the random variable.

For a discrete probability function P(x) ≥ 0 and ∑P(x) = 1 can be given by f(x) = 1/n where n is the number of values the random variable can assume to have.

Cumulative distribution function of a discrete random variable X is denotes as F(x) such that: $F(x)=P(X\le x)=\sum_{x_{i}\le x}{f(x_{i})}$

where 0 ≤ F(x) ≤ 1)

Example: Let X denote the number of hours you play video games during a randomly selected school day. The probability that X can take the values x, where k is some unknown constant is given below:

$P(X=x)=\bigg\{ \begin{gathered} 0.1 if x=0 \\ kx, if x=1 or 2 \\ k(5-x), if x=3 or 4 \\ 0, otherwise \\ \end{gathered}$

From the conditions above, find k. What is the value of probability that you play at least 2 hours? Exactly 2 hours? At most two hours?

Solution: We can express the problem in tabular form to show probability distribution of X:

 X 0 1 2 3 4 5 P(X) 0.1 k 2k 2k k 0

∑P(x) = 1

Therefore,
0.1 + k + 2k + 2k + k = 1

6k = 0.9

k = 0.15

P (Play at least 2 hours ) = P (X ≥ 2) = P(X = 2) + P(X=3) + P(X=4)

= 2k +2k +k + 0 = 5k = 0.75

P (play 2 hours) = P(X = 2)= 2k = 0.3

P (play at most 2 hours) = P(X≤2) = P(X=0) +P(X=1) + P(X = 2)

= 0.1+k+2k = 0.1+3k = 0.55