Lesson 3: Discrete Random Variable (Mean and Variance)

Discrete Random Variable

  • Definition: A discrete random variable is one that can take on a countable number of distinct values.

Mean and Variance

  • Mean (Expected Value): Represents the average outcome of a probability distribution.

  • Variance: Measures the spread of the random variable's values around the mean.

Steps in Computing the Mean of the Probability Distribution

  1. Construct the Probability Distribution

    • Identify the random variable X and list corresponding probabilities P(X).

  2. Multiply Values by Probabilities

    • For each value of X, calculate the product of X and its corresponding P(X).

  3. Add the Results

    • Sum all the products obtained to get the mean.

Example No. 1: Number of Spots from Rolling a Die

  • Outcome Analysis:

    • Number of Spots X: 1, 2, 3, 4, 5, 6

    • Probability P(X): 1/6 for each value (since a die has 6 faces)

  • Calculations:

    • X.P(X) for each X:

      • 1 *(1/6) = 1/6

      • 2 *(1/6) = 2/6

      • 3 *(1/6) = 3/6

      • 4 *(1/6) = 4/6

      • 5 *(1/6) = 5/6

      • 6 *(1/6) = 6/6

    • Sum: ΣX.P(X) = (1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6) = 3.5

Mean Calculation Formula for Probability Distribution

  • Formula: ( E(X) = \sum (X_i \cdot P(X_i)) )

    • Where ( X_i ) are the values of X and ( P(X_i) ) are the corresponding probabilities.

Steps in Computing the Variance of the Probability Distribution

  1. Calculate the Mean: Use the mean formula from above.

  2. Substitute in the Variance Formula:

    • Formula: ( \sigma^2 = \sum (X_i - \mu)^2 \cdot P(X_i) )

  3. Standard Deviation:

    • Calculate the square root of the variance: ( \sigma = \sqrt{\sigma^2} )

Example No. 2: Number of Cars Sold

  • Data on Cars Sold per Day:

    • Number of Cars Sold: 0, 1, 2, 3, 4

    • Corresponding Probability P(X): ( \frac{1}{10}, \frac{2}{10}, \frac{3}{10}, \frac{2}{10}, \frac{2}{10} )

  • Task: Compute variance and standard deviation using provided steps.

Exercise: Admissions Inquiries

  • Data Provided:

    • Number of Inquiries: 22, 23, 24, 25, 26, 27

    • Corresponding Probability P(X): 0.08, 0.19, 0.36, 0.25, 0.07, 0.05

  • Task: Find the variance and standard deviation.