Discrete Random Variable PMF 📈

A random variable is a mathematical concept employed in probability theory and statistics to model and describe the outcomes of a random process or experiment. It assigns numerical values to potential results of a random event. 📊

Formally, a random variable is defined as a function that maps the outcomes of a random experiment to real numbers. It associates a specific numerical value with each outcome, enabling us to quantify and analyze the uncertainty linked to the experiment. 🧮

There are two primary types of random variables: discrete random variables and continuous random variables. 🎯

Discrete Random Variable: A discrete random variable assumes a countable number of distinct values, often integers or whole numbers. Examples include the number of heads when flipping a coin multiple times or the count of cars passing through an intersection in a given time interval. 🎲

Continuous Random Variable: In contrast, a continuous random variable takes on an uncountable infinite number of possible values within a specific range or interval, typically represented by real numbers. Examples encompass the height of individuals, the time for a machine to complete a task, or the temperature at a specific location. 📏

Random variables are foundational in probability theory and statistical analysis, allowing us to define probability distributions and compute probabilities, expected values, variances, and other statistical characteristics associated with the random process or experiment. 📈

Now, let's outline the steps for illustrating the probability distribution of a discrete random variable. 📊

1. Identify the Independent Variable: Begin by identifying the independent variable, denoted as 'x,' in the function y = f(x). In this example, we'll use the number of girls in a family with 3 children as our independent variable.

2. Determine the Sample Space: List all possible outcomes, creating the sample space, which in this case is S = {ggg, ggb, gbg, bgg, bbg, bgb, gbb, bbb}.

3. Assign Probabilities: Assign probabilities to each outcome. For instance:

- Probability of no girls (X = 0) is 1/8.

- Probability of one girl (X = 1) is 3/8 (outcomes: bbg, bgb, gbb).

- Probability of two girls (X = 2) is 3/8 (outcomes: ggb, gbg, bgg).

- Probability of three girls (X = 3) is 1/8.

4. Define the Probability Mass Function (PMF): This function, denoted as P{X = x}, is a mathematical representation of the relationship between X (number of girls) and the corresponding probabilities. We'll call it f(X):

- P{X = x} = f(X).

5. Main Properties of PMF:

- 0 <= X <= 3

- Sum of all f(X) values equals 1 (normalization condition):

- f(0) + f(1) + f(2) + f(3) = 1

6. Graphical Representation: Represent f(X) graphically in the form of a histogram.

7. Calculate the Mean (Expected Value):

- μ ≡ E(X) = ∑X × f(X) = 0 × f(0) + 1 × f(1) + 2 × f(2) + 3 × f(3)

8. Compute the Variance:

- σ^2 = Σ(X - μ)^2 × f(X) ≡ ∑X^2 × f(X) − μ^2

This process helps visualize and analyze the probability distribution of a discrete random variable, providing insights into its central tendencies and variability. 📈📚