Consider the negative binomial distribution given in...

Consider the negative binomial distribution given in Definition $3.9 .$ a. Show that if $y \geq r+1, \frac{p(y)}{p(y-1)}=\left(\frac{y-1}{y-r}\right) q .$ This establishes a recursive relationship between successive negative binomial probabilities, because $p(y)=p(y-1) \times\left(\frac{y-1}{y-r}\right) q$ b. Show that $\frac{p(y)}{p(y-1)}=\left(\frac{y-1}{y-r}\right) q>1$ if $y<\frac{r-q}{1-q} .$ Similarly, $\frac{p(y)}{p(y-1)}<1$ if $y>\frac{r-q}{1-q}$ c. Apply the result in part (b) for the case $r=7, p=.5$ to determine the values of $y$ for which $p(y)>p(y-1)$

Consider the negative binomial distribution given in Definition $3.9 .$
a. Show that if $y \geq r+1, \frac{p(y)}{p(y-1)}=\left(\frac{y-1}{y-r}\right) q .$ This establishes a recursive relationship between successive negative binomial probabilities, because $p(y)=p(y-1) \times\left(\frac{y-1}{y-r}\right) q$
b. Show that $\frac{p(y)}{p(y-1)}=\left(\frac{y-1}{y-r}\right) q>1$ if $y<\frac{r-q}{1-q} .$ Similarly, $\frac{p(y)}{p(y-1)}<1$ if $y>\frac{r-q}{1-q}$
c. Apply the result in part (b) for the case $r=7, p=.5$ to determine the values of $y$ for which $p(y)>p(y-1)$

Key Concepts

Monotonicity and Mode Determination

The monotonicity of the probability mass function is analyzed by investigating when the ratio p(y)/p(y-1) is greater than, equal to, or less than 1. This approach helps in locating the mode, or the most probable value, of the distribution. Specifically, determining the range of y for which the ratio exceeds 1 identifies the region where the probabilities are increasing, and beyond this point, the probabilities begin to decrease, indicating the peak of the distribution.

Successive Ratio of Probabilities

The ratio of successive probabilities (p(y)/p(y-1)) is a central concept in analyzing the shape of a probability mass function. By studying this ratio, one can determine whether the sequence of probabilities is increasing or decreasing with respect to the variable. A ratio greater than 1 indicates that the probabilities are increasing at that point, while a ratio less than 1 implies they are decreasing.

Recursive Probability Computation

A recursive probability computation involves expressing the probability of an event at a certain value in terms of the probability at a previous or lower value. In the context of the negative binomial distribution, this is achieved by relating p(y) to p(y-1) through a ratio that includes parameters and index-dependent factors, simplifying the calculation of probabilities incrementally.

Negative Binomial Distribution

The negative binomial distribution is a discrete probability distribution that models the number of failures before a fixed number of successes occur in a sequence of independent and identically distributed Bernoulli trials. It is commonly used in scenarios where the focus is on the count of unsuccessful outcomes prior to achieving a predetermined number of successful events.

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