Consider Example 3.1.5. (a) For what values of k does the...

Consider Example 3.1.5. (a) For what values of $k$ does the probability mass function of $X$ assume its maximum value? Calculate the value. (b) Using the APL function POISSON $\triangle$ DIST, it was shown that $P[X \leq 15]=0.95126$. Estimate this value using the one-sided inequality (Theorem 2.10.3). (c) The APL function POISSON $\triangle$ DIST shows that $P[4 \leq X \leq$ $16]=0.96262$. Estimate this value using (i) the Chebyshev inequality, and (ii) the normal approximation.

Consider Example 3.1.5.
(a) For what values of $k$ does the probability mass function of $X$ assume its maximum value? Calculate the value.
(b) Using the APL function POISSON $\triangle$ DIST, it was shown that $P[X \leq 15]=0.95126$. Estimate this value using the one-sided inequality (Theorem 2.10.3).
(c) The APL function POISSON $\triangle$ DIST shows that $P[4 \leq X \leq$ $16]=0.96262$. Estimate this value using
(i) the Chebyshev inequality, and
(ii) the normal approximation.

Key Concepts

Chebyshev's Inequality

Chebyshev's inequality is a fundamental result in probability theory that bounds the probability that the value of a random variable with finite mean and variance deviates from its mean by more than a specified number of standard deviations. It applies irrespective of the underlying distribution and provides a conservative estimate that guarantees the fraction of values lying within a given interval around the mean. This inequality is a key tool for assessing variability and tail risk without assuming a specific distribution shape.

One-Sided Inequalities

One-sided inequalities, such as Cantelli's inequality, provide bounds for the probability that a random variable deviates from its mean in one specified direction. Unlike two-sided bounds like Chebyshev’s, these inequalities are used to estimate the probability of events occurring on a one side (either the upper or lower tail) of the distribution. They are particularly useful when only one tail probability is of interest in a probabilistic analysis.

Normal Approximation

The normal approximation involves using the normal distribution to approximate the behavior of another distribution under certain conditions, often when the sample size or the rate parameter in a Poisson distribution is large. For the Poisson distribution, when the mean (and variance) is sufficiently large, the distribution can be approximated by a normal distribution with the same mean and variance. This approximation simplifies calculations for probabilities and confidence intervals, and is widely used in statistical applications where exact computations using the Poisson model are cumbersome.

Poisson Distribution

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, provided these events happen with a known constant mean rate and independently of the time since the last event. Its key property is that both its mean and variance are equal to the rate parameter. This distribution is commonly used to model count data and rare events.

Mode of a Poisson Distribution

The mode of a probability distribution is the value at which its probability mass function (pmf) attains its maximum value. For the Poisson distribution, the mode is typically the greatest integer less than or equal to the rate parameter. In some cases, particularly when the rate parameter is an integer, the pmf may achieve its maximum at more than one value. Understanding the mode helps in identifying the most likely number of occurrences (or the peak of the pmf).

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