Calculating the Expectation of a Discrete Random Variable Using Its Probability Mass Function

This article explains the concept of the probability mass function (PMF) in the context of a discrete random variable and demonstrates how to calculate its expectation. We will follow a step-by-step approach using a practical example to ensure clarity.

Understanding the Probability Mass Function (PMF)

A probability mass function (PMF) defines the probability that a discrete random variable X takes on a specific value. In this article, we will focus on the PMF given by:

For x 1, 2, 3, 4, 5, 6, PX kx.

This implies that the probability for each value of X is proportional to that value itself.

Step-by-Step Calculation

To calculate the constant k, we use the fact that the sum of all probabilities must equal 1:

Solving for k:[21k 1 implies k frac{1}{21}]

Calculating the Expectation

The mean value (or expectation) of the random variable X, denoted by E[X], is given by the weighted sum of the possible values of X, where the weights are the respective probabilities PX:[mathbb{E}X sum_{x1}^{5} x cdot kx]

Substituting k frac{1}{21} and evaluating the sum:[mathbb{E}X sum_{x1}^{5} x cdot frac{1}{21}x frac{1}{21} sum_{x1}^{5} x^2]

The sum of squares of the first 5 natural numbers is:[sum_{x1}^{5} x^2 1^2 2^2 3^2 4^2 5^2 55]

Therefore:[mathbb{E}X frac{1}{21} cdot 55 frac{55}{21} approx 2.619]

Conclusion

In conclusion, the expectation of the given probability mass function is approximately 2.619, indicating that the random variable X is centered around this value when considering the product of its values and their respective probabilities.

Understanding the calculation of the expectation using the PMF is fundamental to analyzing discrete random variables and making predictions based on their behavior.

Related Keywords

- probability mass function (PMF)

- discrete random variable

- expectation (mathematical expectation)