Problem 1 - Basic Statistics for Business & Economics

Question

Compute the mean and variance of the following discrete probability distribution. $$ \begin{array}{|cc|} \hline x & P(x) \\ \hline 0 & .2 \\ 1 & .4 \\ 2 & .3 \\ 3 & .1 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The mean is 1.3 and the variance is 0.81.

Step 1: Identify Values and Probabilities

Identify the random variable values and their corresponding probabilities from the table given: \[ x = \{0, 1, 2, 3\}, \ P(x) = \{0.2, 0.4, 0.3, 0.1\} \].

Step 2: Compute the Mean (Expected Value)

Calculate the mean (expected value) of the distribution using the formula \( E(X) = \sum (x_i \cdot P(x_i)) \), where \( x_i \) are the values of the random variable and \( P(x_i) \) are their probabilities. \[ E(X) = (0 \cdot 0.2) + (1 \cdot 0.4) + (2 \cdot 0.3) + (3 \cdot 0.1) = 0 + 0.4 + 0.6 + 0.3 = 1.3 \].

Step 3: Compute E(X^2)

Calculate \( E(X^2) \) which is needed for the variance. Use the formula \( E(X^2) = \sum (x_i^2 \cdot P(x_i)) \). \[ E(X^2) = (0^2 \cdot 0.2) + (1^2 \cdot 0.4) + (2^2 \cdot 0.3) + (3^2 \cdot 0.1) = 0 + 0.4 + 1.2 + 0.9 = 2.5 \].

Step 4: Compute the Variance

Calculate the variance using the formula \( Var(X) = E(X^2) - [E(X)]^2 \). \[ Var(X) = 2.5 - (1.3)^2 = 2.5 - 1.69 = 0.81 \].

Key Concepts

Mean Calculation

To find the mean of a discrete probability distribution, you need to determine what is commonly referred to as the 'Expected Value' (E(X)). This is essentially a measure of the center of the distribution, providing a sense of the average outcome you can expect.

The formula for the mean in a probability distribution is:

• \( E(X) = \sum (x_i \cdot P(x_i)) \)

Where:

• \( x_i \) are the values of the random variable
• \( P(x_i) \) are the probabilities associated with these values

To illustrate, using the provided data:

• \( E(X) = (0 \cdot 0.2) + (1 \cdot 0.4) + (2 \cdot 0.3) + (3 \cdot 0.1) \)
• This simplifies to \( 0 + 0.4 + 0.6 + 0.3 = 1.3 \)

Thus, the mean or expected value is 1.3, providing us a point of reference for what is typical in this distribution.

Variance Calculation

Variance is a crucial statistical measure that tells you how much the values in your distribution vary or spread out from the mean. It helps to understand the degree of uncertainty or risk associated with a random variable.

The variance \( Var(X) \) is calculated with this formula:

• \( Var(X) = E(X^2) - [E(X)]^2 \)

Initially, we need to find \( E(X^2) \), which represents the expected value of the squares of the random variable. The calculation for \( E(X^2) \) uses:

• \( E(X^2) = \sum (x_i^2 \cdot P(x_i)) \)

From the given example:

• \( E(X^2) = (0^2 \cdot 0.2) + (1^2 \cdot 0.4) + (2^2 \cdot 0.3) + (3^2 \cdot 0.1) \)
• This equals \( 0 + 0.4 + 1.2 + 0.9 = 2.5 \)

Now, we calculate variance as follows:

• \( Var(X) = 2.5 - (1.3)^2 \)
• Which results in \( 2.5 - 1.69 = 0.81 \)

Thus, the variance of this distribution is 0.81, indicating the average degree to which each value differs from the mean.

Expected Value

The expected value is a fundamental concept that gives an average of all possible outcomes in a probability distribution, weighted by their respective probabilities. It acts as a benchmark for making decisions in a probabilistic framework by predicting what happens on average over many trials.

In simpler terms, it tells you the expected result when an experiment is repeated many times. This makes it incredibly useful in real-life scenarios such as games of chance, insurance modeling, and various fields of risk management.

The steps to calculate the expected value, as outlined earlier, involve multiplying each possible outcome by its probability and summing the results. Here's a quick recap:

• Identify each outcome \( x_i \) and its probability \( P(x_i) \).
• Calculate \( E(X) = \sum (x_i \cdot P(x_i)) \).

By giving you insights into the likely average outcome, the expected value serves as a powerful tool to steer decisions and predictions.