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