Random Variable and its Probability Distribution A random variable is a variable whose value is uncertain and can take on different values. It is typically r...
A random variable is a variable whose value is uncertain and can take on different values. It is typically represented by a letter, such as X, Y, or Z.
The probability distribution of a random variable is a mathematical function that describes the probability of the variable taking different values. It is expressed in the form of a probability mass function (PMF) or a probability density function (PDF).
Probability Mass Function (PMF)
A PMF gives the probability of a specific discrete value being taken by the random variable. It can be represented by a table or a graph that shows the probability of each possible outcome.
Probability Density Function (PDF)
A PDF gives the probability of a continuous random variable taking a specific range of values. It can be represented by a graph that shows the probability density at different values in the range.
The probability distribution helps us understand the likelihood of different outcomes when we have a random variable. By knowing the probability distribution, we can make predictions about the possible values of the random variable and calculate the expected value.
Examples
A roll of a fair six-sided die. The probability distribution for this random variable would be a discrete PMF with outcomes from 1 to 6.
The amount of time it takes for a light bulb to burn out. The probability distribution for this random variable would be a continuous PDF with a higher probability for shorter durations.
The number of customers that visit a store in a given day. The probability distribution for this random variable would be a discrete PMF with outcomes representing the number of customers