Problem 65 - A First Course in Probability

Question

The number of winter storms in a good year is a Poisson random variable with mean \(3,\) whereas the number in a bad year is a Poisson random variable with mean \(5 .\) If next year will be a good year with probability .4 or a bad year with probability.6, find the expected value and variance of the number of storms that will occur.

Step-by-Step Solution

Answer

The expected number of storms next year is 4.2, and the variance of the number of storms next year is 19.56.

Step 1: Compute the expected value and variance for each condition separately

For a good year, the mean of winter storms (given as \(\lambda_{good}\)) is 3. Since we know that the expected value of a Poisson random variable is equal to its mean: \(E(X_{good}) = \lambda_{good} = 3\) The variance for a Poisson random variable is also equal to its mean: \(Var(X_{good}) = \lambda_{good} = 3\) For a bad year, the mean of winter storms (given as \(\lambda_{bad}\)) is 5. Similar to above, \(E(X_{bad}) = \lambda_{bad} = 5\) \(Var(X_{bad}) = \lambda_{bad} = 5\)

Step 2: Apply the law of total expectation

The law of total expectation states that the expected value of a random variable can be found by taking the weighted average of the expected values of its conditional probabilities. In this case, we will apply the law of total expectation to find the overall expected number of storms next year: \(E(X) = E(X_{good}) * P(Good\;Year) + E(X_{bad}) * P(Bad\;Year)\) Substituting the values from Step 1 and probabilities from the problem statement: \(E(X) = 3 * 0.4 + 5 * 0.6 = 1.2 + 3 = 4.2\) Therefore, the expected number of storms next year is 4.2.

Step 3: Apply the law of total variance

The law of total variance states that the variance of a random variable can be found by taking the weighted average of the variances and the expected values of its conditional probabilities squared. In this case, we will apply the law of total variance to find the overall variance of the number of storms next year: \(Var(X) = E[Var(X|Y)] + Var[E(X|Y)]\) Here, \(Y\) refers to whether the year is a good or bad year. We can rewrite this expression using the given probabilities: \(Var(X) = [Var(X_{good}) * P(Good\;Year) + Var(X_{bad}) * P(Bad\;Year)] + [E(X_{good}^2) * P(Good\;Year) + E(X_{bad}^2) * P(Bad\;Year) - E(X)^2]\) Substituting the values from Step 1 and the problem statement: \(Var(X) = [3*0.4 + 5*0.6] + [(3 + 9)*0.4 + (5 + 25)*0.6 - 4.2^2]\) \(Var(X) = 3.6 + 33.6 - 17.64 = 19.56\) Therefore, the variance of the number of storms next year is 19.56.

Key Concepts

Expected Value

The expected value is one of the fundamental concepts in probability and statistics, representing the average outcome you would anticipate after many repetitions of an experiment or random event. When dealing with a Poisson random variable, which is used for modeling the number of times an event happens in a fixed interval or space, the expected value is particularly straightforward: it is equal to the mean, denoted by \(\lambda\).
For instance, in our exercise regarding the number of winter storms, the expected value for a good year with a mean of 3 is just \(E(X_{good}) = \lambda_{good} = 3\). Similarly, for a bad year with a mean of 5, the expected value is \(E(X_{bad}) = \lambda_{bad} = 5\). This simple principle helps students to quickly determine the central tendency of events described by the Poisson distribution.

Variance

Variance is a measure of how spread out numbers are in a data set, indicating the degree of variability around the expected value. For a Poisson random variable, the beauty lies in its simplicity: the variance is always equal to its mean \(\lambda\).

This ties directly into the problem at hand, where the variance for the number of winter storms in a good year (\