Problem 66 - A First Course in Probability

Question

In Example \(5 \mathrm{c},\) compute the variance of the length of time until the miner reaches safety.

Step-by-Step Solution

Answer

The variance of the length of time until the miner reaches safety, denoted as \(Var(X)\), can be calculated using the formula: \(Var(X) = \sum_{i=1}^{n} (x_i - E(X))^2 P(x_i)\) Here, \(E(X)\) is the expected value of the random variable X, and \(P(x_i)\) represents the probability distribution of X. Make sure to use the given probability distribution from Example 5c to perform the calculations and find the final answer for the variance.

Step 1: Define the Random Variable

Let X be the random variable representing the length of time until the miner reaches safety. We will be working with this random variable throughout the exercise.

Step 2: Determine the Probability Distribution of X

Based on the information given in Example 5c, we can determine the probability distribution of X (for example, the chances of reaching safety in 1 hour, 2 hours, 3 hours, etc.). Make sure to reference the given problem's probability distribution for this step.

Step 3: Calculate the Expected Value (Mean) of X

To find the expected value of X, we multiply each possible time value until the miner reaches safety by its respective probability and sum them up. The formula for expected value (mean) is: \(E(X) = \sum_{i=1}^{n} x_i P(x_i)\) Where \(n\) is the total number of possible outcomes, \(x_i\) is the time until the miner reaches safety for the \(i^{th}\) outcome, and \(P(x_i)\) is the probability of the \(i^{th}\) outcome. Find the expected value of X using the probability distribution determined in Step 2.

Step 4: Calculate the Variance of X

To find the variance, we first calculate the squared differences between each time value and the expected value, multiply by the corresponding probabilities, and then sum up these products. The formula for variance is: \(Var(X) = \sum_{i=1}^{n} (x_i - E(X))^2 P(x_i)\) Where \(n\) is the total number of possible outcomes, \(x_i\) is the time until the miner reaches safety for the \(i^{th}\) outcome, \(E(X)\) is the expected value of X, and \(P(x_i)\) is the probability of the \(i^{th}\) outcome. Calculate the variance of X using the expected value and the probability distribution determined in the previous steps.

Step 5: Final Answer

Once you've calculated the variance, write it as the final answer for the length of time until the miner reaches safety.