Suppose X is a Poisson random variable with E[X]=α. (a) Show that P[X=k] ≥P[X=k+1] implies that

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Suppose X is a Poisson random variable with E[X]=α. (a) Show...

Suppose $X$ is a Poisson random variable with $E[X]=\alpha$. (a) Show that $P[X=k] \geq P[X=k+1]$ implies that $k+1 \geq \alpha$ and conversely. (b) Show that $P[X=k-1] \leq P[X=k]$ implies that $k \leq \alpha$ and conversely. (c) Use (a) and (b) to show that the pmf of $X, P[X=k]$ first increases monotonically, then decreases monotonically, reaching its greatest value when $\alpha-1 \leq k \leq \alpha$. For example, if $\alpha=4$, then the maximum values of the pmf occur at $k=3,4$. The values are $$ e^{-4} \frac{4^3}{3 !}=e^{-4} \frac{4^4}{4 !}=0.19537 . $$

Suppose $X$ is a Poisson random variable with $E[X]=\alpha$.
(a) Show that $P[X=k] \geq P[X=k+1]$ implies that $k+1 \geq \alpha$ and conversely.
(b) Show that $P[X=k-1] \leq P[X=k]$ implies that $k \leq \alpha$ and conversely.
(c) Use (a) and (b) to show that the pmf of $X, P[X=k]$ first increases monotonically, then decreases monotonically, reaching its greatest value when $\alpha-1 \leq k \leq \alpha$. For example, if $\alpha=4$, then the maximum values of the pmf occur at $k=3,4$. The values are
$$
e^{-4} \frac{4^3}{3 !}=e^{-4} \frac{4^4}{4 !}=0.19537 .
$$

Key Concepts

Unimodality and Mode Determination

Unimodality refers to the property of a distribution to have a single mode, meaning the pmf increases to a peak and then decreases. For the Poisson distribution, by analyzing the ratio of consecutive probabilities, it is shown that the distribution's pmf increases monotonically until a certain point when k is less than or equal to ?, and decreases afterward. This analysis demonstrates that the mode, or the point of maximum probability, occurs when the integer k is near the value of ?, specifically in the interval [??1, ?].

Ratio of Consecutive Probabilities

Comparing consecutive probabilities, specifically the ratio P[X = k+1] / P[X = k], is a technique used to analyze the behavior of the pmf. Establishing whether this ratio is less than, equal to, or greater than one helps in determining whether the pmf is increasing or decreasing at a given point, which in turn provides insights into the location of the maximum probability (mode) of the distribution.

Poisson Distribution

The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval of time or space when these events occur with a known constant rate independently of the time since the last event. It is characterized by its parameter ?, which represents both its mean and variance, and its probability mass function (pmf) is given by P[X = k] = e^(–?) (?^k)/(k!).

Probability Mass Function (pmf)

The probability mass function (pmf) describes the probability that a discrete random variable is exactly equal to some value. For the Poisson distribution, the pmf is used to determine the likelihood of observing k events, and the behavior of this function (increasing, peaking, and then decreasing) is crucial for understanding the distribution's shape and mode.

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