Question

Let \(X\) have a Poisson distribution with \(\mu=100\). Use Chebyshev's inequality to determine a lower bound for \(P(75

Short answer

The lower bound for \(P(75<X<125)\) according to Chebyshev's inequality is \(0.84\). The exact probability should be calculated using R, and then it should be compared with the theoretical result for evaluating the accuracy of Chebyshev's inequality.

Step-by-step solution

Understanding Chebyshev's Inequality

Chebyshev's inequality states that \(P(\mu - k\sigma < X < \mu + k\sigma) \geq 1 - \frac{1}{k^2}\) where \(\mu\) is the mean, and \(\sigma\) is the standard deviation. Here \(\mu = 100\) as given. For a Poisson distribution, both the mean and the variance are equal to \(\mu\), so \(\sigma = \sqrt{\mu} = \sqrt{100} = 10\). The values to insert into the inequality are \(75\) and \(125\) for \(X\). Calculate \(k1 = \frac{|\mu - 75|}{\sigma} = 2.5\) and \(k2 = \frac{|\mu - 125|}{\sigma} = 2.5\). Since \(k1 = k2\), we choose the largest \(k\) to get the most information, in this case both are \(2.5\). So, according to Chebyshev's inequality, \(P(75 < X < 125) \geq 1 - \frac{1}{k^2} = 1 - \frac{1}{2.5^2} = 0.84\).

Calculate Probability Using R

Now use R to calculate the exact value. The corresponding R command is ppois(125, lambda=100) - ppois(74, lambda=100). This will give the exact probability. The ppois() function in R gives the probability that a Poisson random variable is less than or equal to a certain value, so we subtract the two cumulative probabilities to get the probability within the range.

Compare the Theoretical and Empirical Probabilities

Compare the value you got from Chebyshev's inequality (Step 1) and from R (Step 2). If they are close, then the approximation by Chebyshev's inequality is accurate. Note that since Chebyshev's inequality gives a bound, not an exact equation, the result from R should be less than or equal to the Chebyshev result.