We prove that if Z is a Poisson random variable of mean μ, where μ≥1 is an integer, then Pr(Z ≥μ

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We prove that if Z is a Poisson random variable of mean μ,...

We prove that if $Z$ is a Poisson random variable of mean $\mu$, where $\mu \geq$ 1 is an integer, then $\operatorname{Pr}(Z \geq \mu) \geq 1 / 2$ and $\operatorname{Pr}(Z \leq \mu) \geq 1 / 2$. (a) Show that $\operatorname{Pr}(Z=\mu+h) \geq \operatorname{Pr}(Z=\mu-h-1)$ for $0 \leq h \leq \mu-1$. (b) Using part (a), argue that $\operatorname{Pr}(Z \geq \mu) \geq 1 / 2$. (c) Show that $\operatorname{Pr}(Z=\mu-h) \geq \operatorname{Pr}(Z=\mu+h+1)$ for $0 \leq h \leq \mu$. (d) Determine a lower bound on $\operatorname{Pr}(Z=\mu-h)-\operatorname{Pr}(Z=\mu+h+1)$. (e) Determine an upper bound on $\operatorname{Pr}(Z \geq 2 \mu+2)$. (f) Using parts (c) $-(\mathrm{e})$, argue that $\operatorname{Pr}(Z \leq \mu) \geq 1 / 2$.

We prove that if $Z$ is a Poisson random variable of mean $\mu$, where $\mu \geq$ 1 is an integer, then $\operatorname{Pr}(Z \geq \mu) \geq 1 / 2$ and $\operatorname{Pr}(Z \leq \mu) \geq 1 / 2$.
(a) Show that $\operatorname{Pr}(Z=\mu+h) \geq \operatorname{Pr}(Z=\mu-h-1)$ for $0 \leq h \leq \mu-1$.
(b) Using part (a), argue that $\operatorname{Pr}(Z \geq \mu) \geq 1 / 2$.
(c) Show that $\operatorname{Pr}(Z=\mu-h) \geq \operatorname{Pr}(Z=\mu+h+1)$ for $0 \leq h \leq \mu$.
(d) Determine a lower bound on $\operatorname{Pr}(Z=\mu-h)-\operatorname{Pr}(Z=\mu+h+1)$.
(e) Determine an upper bound on $\operatorname{Pr}(Z \geq 2 \mu+2)$.
(f) Using parts (c) $-(\mathrm{e})$, argue that $\operatorname{Pr}(Z \leq \mu) \geq 1 / 2$.

Key Concepts

Symmetry and Comparison Arguments

Although the Poisson distribution is not symmetric, comparison arguments can be constructed by comparing probabilities at equidistant points from the mean. By showing that the probability mass at positions above the mean is at least as large as corresponding positions below the mean (and vice versa), one can argue effectively about the balance between the two tails. This insight forms the basis of proofs that each tail (i.e., values below and above the mean) contributes at least 50% of the cumulative probability.

Tail Probability Inequalities

Tail probability inequalities provide bounds on the probabilities of the events lying in the tails of a distribution, meaning events significantly above or below the mean. In the context of the Poisson distribution, showing that the probabilities on either side of the mean are at least 1/2 involves comparing the probabilities in the upper and lower tails. Such techniques are crucial in obtaining lower and upper bounds for cumulative distribution function segments of discrete random variables.

Poisson Distribution

A Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval, with these events happening with a known constant mean rate and independently of the time since the last event. Its probability mass function is defined as P(Z=k) = e^(–?) ?^k/k!, where ? is both the mean and the variance. This distribution often models count data and is critical in understanding inequalities around its mean.

Recurrence Relations in Discrete Probability

Recurrence relations describe the relationship between probabilities at adjacent or symmetric points in a discrete distribution. For a Poisson random variable, using the fact that P(Z=k+1)/P(Z=k) = ?/(k+1) allows one to compare probabilities at different points. This comparison is fundamental when establishing inequalities between probabilities at symmetric distances from the mean and is used in proving bounds on cumulative probabilities.

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