Let p(y) denote the probability function associated with a...

Let $p(y)$ denote the probability function associated with a Poisson random variable with mean $\lambda$ a. Show that the ratio of successive probabilities satisfies $\frac{p(y)}{p(y-1)}=\frac{\lambda}{y},$ for $y=1,2, \ldots$ b. For which values of $y$ is $p(y) > p(y-1) ?$ c. Notice that the result in part (a) implies that Poisson probabilities increase for awhile as $y$ increases and decrease thereafter. Show that $p(y)$ maximized when $y=$ the greatest integer less than or equal to $\lambda$

Let $p(y)$ denote the probability function associated with a Poisson random variable with mean $\lambda$
a. Show that the ratio of successive probabilities satisfies $\frac{p(y)}{p(y-1)}=\frac{\lambda}{y},$ for $y=1,2, \ldots$
b. For which values of $y$ is $p(y) > p(y-1) ?$
c. Notice that the result in part (a) implies that Poisson probabilities increase for awhile as $y$ increases and decrease thereafter. Show that $p(y)$ maximized when
$y=$ the greatest integer less than or equal to $\lambda$

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Key Concepts

Mode of a Discrete Distribution

The mode is the value at which the probability mass function reaches its maximum. In the context of the Poisson distribution, examining the ratio of successive probabilities helps identify the point where the PMF transitions from increasing to decreasing, thereby determining the mode of the distribution.

Successive Probability Ratios

The ratio of successive probabilities, derived by dividing p(y) by p(y – 1), offers insight into the behavior of the distribution. For the Poisson distribution, this ratio simplifies to ?/y, which is central for determining whether the probabilities increase or decrease as y increments.

Poisson Distribution

This is a discrete probability distribution used to model the number of events occurring in a fixed interval of time or space, under the assumption that these events occur with a known constant mean rate and independently of the time since the last event.

Probability Mass Function (PMF)

The PMF of a discrete random variable assigns probabilities to each possible outcome. For the Poisson distribution, the PMF is expressed as p(y) = (e^(–?) * ?^y) / y!, providing the probability of y events occurring within the interval.

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