Let X be a Poisson random variable with parameter α. Prove that (a) P[X ≤α/ 2] ≤4 /(α+4)<4 / αan

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Let X be a Poisson random variable with parameter α. Prove...

Let $X$ be a Poisson random variable with parameter $\alpha$. Prove that (a) $P[X \leq \alpha / 2] \leq 4 /(\alpha+4)<4 / \alpha$ and (b) $P[X \geq 2 \alpha] \leq 1 /(1+\alpha)<1 / \alpha$. (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 $$ e^{-4} \frac{4^3}{3 !}=e^{-4} \frac{4^4}{4 !}=0.19537 . $$

Let $X$ be a Poisson random variable with parameter $\alpha$. Prove that
(a) $P[X \leq \alpha / 2] \leq 4 /(\alpha+4)<4 / \alpha$ and
(b) $P[X \geq 2 \alpha] \leq 1 /(1+\alpha)<1 / \alpha$.
(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

Poisson Distribution

The Poisson distribution is used to model the number of events occurring in a fixed interval of time or space when these events happen with a known constant mean rate and independently of the time since the last event. It is characterized by its parameter ?, which represents both the mean and variance of the distribution. Understanding the Poisson distribution is fundamental since it is widely used in various areas such as queueing theory, reliability engineering, and event counting in natural phenomena.

Probability Mass Function (pmf)

The probability mass function (pmf) for a discrete random variable provides the probability that the variable takes a specific value. For a Poisson random variable, the pmf is given by P[X = k] = e^(-?) ?^k / k!, where k is a non-negative integer. This function is key to analyzing the distribution’s behavior, including its mean, variance, and the locations of its maximum values (modes).

Monotonicity and Mode of the pmf

Monotonicity in this context refers to the behavior of the pmf as the outcome k increases. For a Poisson distribution, the pmf first increases and then decreases, reaching its maximum (the mode) when k is near the parameter value ?. By comparing consecutive values P[X = k] and P[X = k + 1], one can determine where the transition from increasing to decreasing occurs, which is crucial for identifying the mode and understanding the overall shape of the distribution.

Inequalities and Tail Bounds

Inequalities and tail bounds play a significant role in quantifying the probabilities that a random variable deviates significantly from its mean. In the case of the Poisson distribution, establishing bounds for events such as X being less than or equal to ?/2 (lower tail) or greater than or equal to 2? (upper tail) helps in understanding the concentration of the distribution's mass. These bounds are important in practical applications where approximations and risk assessments require knowledge about the likelihood of extreme deviations.

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