Problem 4 - A First Course in Probability

Question

Let \(X_{1}, \ldots, X_{20}\) be independent Poisson random variablcs with mean 1. (a) Use the Markov incquality to obtain a bound on $$P\left\\{\sum_{1}^{20} X_{i}>15\right\\}$$ (b) Use the central limit theorem to approximate $$P\left\\{\sum_{1}^{20} X_{i}>15\right\\}$$

Step-by-Step Solution

Answer

(a) Using the Markov Inequality, we obtain a bound on the probability as \(\frac{4}{3}\). (b) Using the Central Limit Theorem, we approximate \(P\left(\sum_{1}^{20} X_{i}>15\right)\) to be approximately 0.9522.

Step 1: (a) Using the Markov Inequality

Recall that the Markov Inequality states that for any non-negative random variable \(Y\) and any \(a > 0\), we have: \[P(Y \ge a) \le \frac{E[Y]}{a}\] We have a sum of 20 independent Poisson random variables, each with a mean of 1. Let \(Y = \sum_{i=1}^{20} X_{i}\), so the expected value of \(Y\), \(E[Y] = E\left[\sum_{i=1}^{20} X_{i}\right] = \sum_{i=1}^{20} E[X_{i}] = 20 \times 1 = 20\). Now, let \(a = 15\). Applying the Markov Inequality, we get: \[P\left(\sum_{1}^{20} X_{i}>15\right) = P(Y \ge 15) \le \frac{E[Y]}{15} = \frac{20}{15} = \frac{4}{3}\] So, using the Markov Inequality, we obtain a bound on the probability as \(\frac{4}{3}\).

Step 2: (b) Using the Central Limit Theorem

To use the Central Limit Theorem (CLT), we need to calculate the mean and variance of the sum of the 20 Poisson random variables. Recall that the mean and variance of a Poisson random variable with mean \(\lambda\) are both equal to \(\lambda\). In this problem, each \(X_i\) has a mean of 1, and since they are independent, the variance of each \(X_i\) is also 1. To compute CLT, we need the combined mean and combined variance. The combined mean is given by: \[\mu = \sum_{i=1}^{20} E[X_i] = 20 \times 1 = 20\] The combined variance is given by: \[\sigma^2 = \sum_{i=1}^{20} Var(X_i) = 20 \times 1 = 20\] Now, we can approximate the probability we are interested in as follows: \[P\left(\sum_{1}^{20} X_{i}>15\right) \approx P\left(\frac{\sum_{1}^{20} X_{i} - \mu}{\sqrt{\sigma^2}} > \frac{15 - 20}{\sqrt{20}}\right) = P\left(Z > - \frac{5}{\sqrt{20}}\right)\] Here, Z is a standard normal random variable. Using a standard normal table or calculator, we find that: \[P(Z > -\frac{5}{\sqrt{20}}) \approx 0.9522\] Thus, using the Central Limit Theorem, we approximate \(P\left(\sum_{1}^{20} X_{i}>15\right)\) to be approximately 0.9522.

Key Concepts

Markov Inequality

The Markov Inequality is a handy tool in probability theory giving us a simple upper bound on the probability that a non-negative random variable exceeds a certain value. It's defined for a random variable, say Y, and any positive number a, and tells us that the probability of Y being at least a is at most the expected value of Y divided by a.

In mathematical terms, if Y is non-negative, then:
\[P(Y \ge a) \le \frac{E[Y]}{a}\]
For instance, in the context of our Poisson random variables example, we can use it to bound the probability of their sum exceeding 15. Since their expected sum is 20, the Markov Inequality suggests this probability cannot exceed 20/15 or approximately 1.33—which is a rough bound since it exceeds 1, the maximum probability for any event.

Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental principle in statistics that enables us to predict the behavior of sums of random variables, irrespective of the original distribution, as long as there are sufficient observations. It tells us that the sum (or average) of a large number of independent and identically distributed random variables follows an approximate normal distribution.

In the exercise given, applying the CLT to the sum of 20 Poisson random variables allows us to approximate this sum's distribution with a normal distribution, which is determined by the same mean and variance of the sum. This paves the way for a more accurate probability estimation using the normal distribution, which in this case is about 0.9522 for the given sum exceeding 15.

Probability bounds

Probability bounds give us ways to approximate the likelihood of events, especially when it is complex or impractical to calculate exact probabilities. Tools like Markov inequality provide upper bounds, while the Central Limit Theorem aids in approximating probabilities based on the normal distribution.

In applying such bounds, it's crucial to understand the limitations and conditions under which they can be applied effectively. For instance, the Markov inequality does not require any assumptions about the specific distribution of the random variable, making it widely applicable but less tight, while the CLT provides a better approximation under the condition of a large enough number of observations.

Poisson distribution properties

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known average rate and independently of the time since the last event.

Some critical properties of the Poisson distribution include:

• The mean and variance of a Poisson-distributed random variable are both equal to the parameter λ (lambda).
• Poisson distribution is suitable for modeling the number of events in fixed intervals of time or space when these events happen at a constant rate and independently.
• It is a limiting case of the binomial distribution with the number of trials going to infinity while the expected number of successes remains fixed.

Understanding these properties is essential when we are working with random variables that are thought to be Poisson distributed, as they heavily influence the behavior of sums of such variables and the application of results like the Markov inequality or the CLT.