Problem 33 - Basic Statistics for Business & Economics

Question

An investment will be worth \(\$ 1,000, \$ 2,000,\) or \(\$ 5,000\) at the end of the year. The probabilities of these values are \(.25, .60,\) and \(.15,\) respectively. Determine the mean and variance of the investment's dollar value.

Step-by-Step Solution

Answer

The mean is \( \$2,200 \) and the variance is \( 1,560,000 \).

Step 1: Identify the Possible Outcomes and Probabilities

The investment can result in three different dollar values at year-end: \( \\(1,000 \) with a probability of 0.25, \( \\)2,000 \) with a probability of 0.60, and \( \$5,000 \) with a probability of 0.15.

Step 2: Calculate the Mean of the Investment

The mean (expected value) is calculated using the formula for the expected value of a discrete random variable: \( E(X) = \sum x_i p_i \). Here, we compute: \( E(X) = (1000)(0.25) + (2000)(0.60) + (5000)(0.15) = 250 + 1200 + 750 = 2200 \). Thus, the mean value of the investment is \( \$2,200 \).

Step 3: Calculate Each Squared Deviation from the Mean

We first need to calculate each squared deviation from the mean: \( (1000 - 2200)^2 = 1440000 \), \( (2000 - 2200)^2 = 40000 \), and \( (5000 - 2200)^2 = 7840000 \).

Step 4: Calculate the Variance of the Investment

Variance is the expected value of the squared deviations from the mean. We use the formula \( \text{Var}(X) = \sum (x_i - \mu)^2 p_i \). Calculate:\[ \text{Var}(X) = (1440000)(0.25) + (40000)(0.60) + (7840000)(0.15) = 360000 + 24000 + 1176000 = 1560000 \].Thus, the variance is \( 1,560,000 \).

Key Concepts

Expected Value

The expected value is a fundamental concept in probability and statistics. It represents the average outcome of a random variable over numerous trials. In simpler terms, it is a way to determine what we can expect on average in the long run if an experiment is repeated many times.
For example, if we evaluate an investment, we may want to know the expected profit or value. The expected value helps provide a single summary measure of the random variable's probability distribution.

• The expected value is computed by multiplying each possible outcome by its respective probability.
• Then, all these products are summed up together.

In the provided exercise, the expected value of the investment was calculated using: \[E(X) = (1000 \times 0.25) + (2000 \times 0.60) + (5000 \times 0.15) = 2200\]This means that, over time, the expected average value of the investment is $2,200.

Discrete Random Variable

A discrete random variable is a type of random variable that can take on finite or countable values. These values often come from outcomes of a statistical experiment.
In the context of our exercise, the investment dollar values are examples of a discrete random variable. Each outcome, such as $1,000 or $5,000, is a distinct and countable value that the random variable can assume.

• Discrete random variables are contrasted with continuous random variables, which can take on any value within a given range.
• The probability distribution of a discrete random variable lists all its possible distinct values.

Understanding discrete random variables is important because they form the basis for calculating probabilities and deriving other metrics like expected value and variance.

Probability Distribution

A probability distribution provides a comprehensive view of all possible values a random variable can take, along with their corresponding probabilities.
In a discrete setting, it is a table or a formula that assigns a probability to each possible outcome of a random variable. The total sum of probabilities across all possible values must equal 1.

• In our exercise, the probability distribution of the investment values included three possibilities: $1,000, $2,000, and $5,000.
• Each value is associated with probabilities of 0.25, 0.60, and 0.15, respectively.

This distribution helps us calculate different statistical measures, such as the expected value and variance, by organizing the outcomes into a structured format. Knowing the probability distribution is essential as it enables predictions and estimations based on potential future occurrences.

Statistical Calculations

Statistical calculations are crucial tools in analyzing data and making informed decisions based on random variables and their distributions. This process includes computations like the mean and variance, which provide insights into the central tendency and dispersion of data.

• The **mean**, as described earlier, is the average value expected from a random variable. It offers a quick summary of the data's central point.
• The **variance** measures how much the values of a random variable differ from the mean. A larger variance indicates a wider spread of values.
  It is computed by evaluating each squared deviation from the mean, multiplying it by the probability, and summing these products: \[ \text{Var}(X) = \sum (x_i - \mu)^2 p_i \]

In the exercise, the variance was computed using:\[\text{Var}(X) = (1440000 \times 0.25) + (40000 \times 0.60) + (7840000 \times 0.15) = 1560000\]Understanding these calculations is vital for evaluating the risk and consistency of investments and for broader applications in various data analysis scenarios.