Recall that b(k ; n, p) is the notation for the probability...

Recall that $b(k ; n, p)$ is the notation for the probability that a binomial random variable with parameters $n$ and $p$ assumes the value $k$; that is, $$ b(k ; n, p)=\left(\begin{array}{l} n \\ k \end{array}\right) p^k(1-p)^{n-k} . $$ Consider the sequence $b(0 ; n, p), b(1 ; n, p), \ldots, b(n ; n, p)$. (a) Show that the term $b(k ; n, p)$ is greater than $b(k-1 ; n, p)$ for $1 \leq k<(n+1) p$ and is smaller for $k>(n+1) p$. (b) Show also that, if $(n+1) p=m$ is an integer, then $b(m ; n, p)=$ $b(m-1 ; n, p)$. Note: Since there is exactly one integer $m$ such that $(n+1) p-1<$ $m \leq(n+1) p$, we see by (a) that the maximum value in the sequence is $b(m ; n, p)$. It is called the central term or the "the most probable number of successes". For example, if $n=19$ and $p=0.4$, then $m=8$ and the maximum value of $b(k ; 19,0.4)$ is $$ \begin{aligned} b(8 ; 19,0.4) & =\left(\begin{array}{c} 19 \\ 8 \end{array}\right) 0.4^8 0.6^{11} \\ & =0.179705788 \\ & =\left(\begin{array}{c} 19 \\ 7 \end{array}\right) 0.4^7 0.6^{12} \\ & =b(7 ; 19,0.4) . \end{aligned} $$

Recall that $b(k ; n, p)$ is the notation for the probability that a binomial random variable with parameters $n$ and $p$ assumes the value $k$; that is,
$$
b(k ; n, p)=\left(\begin{array}{l}
n \\
k
\end{array}\right) p^k(1-p)^{n-k} .
$$
Consider the sequence $b(0 ; n, p), b(1 ; n, p), \ldots, b(n ; n, p)$.
(a) Show that the term $b(k ; n, p)$ is greater than $b(k-1 ; n, p)$ for $1 \leq k<(n+1) p$ and is smaller for $k>(n+1) p$.
(b) Show also that, if $(n+1) p=m$ is an integer, then $b(m ; n, p)=$ $b(m-1 ; n, p)$.
Note: Since there is exactly one integer $m$ such that $(n+1) p-1<$ $m \leq(n+1) p$, we see by (a) that the maximum value in the sequence is $b(m ; n, p)$. It is called the central term or the "the most probable number of successes". For example, if $n=19$ and $p=0.4$, then $m=8$ and the maximum value of $b(k ; 19,0.4)$ is
$$
\begin{aligned}
b(8 ; 19,0.4) & =\left(\begin{array}{c}
19 \\
8
\end{array}\right) 0.4^8 0.6^{11} \\
& =0.179705788 \\
& =\left(\begin{array}{c}
19 \\
7
\end{array}\right) 0.4^7 0.6^{12} \\
& =b(7 ; 19,0.4) .
\end{aligned}
$$

Key Concepts

Binomial Distribution

This is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. Its probability mass function is given by a combinatorial coefficient multiplied by appropriate powers of the success probability and the complement probability, reflecting the different ways successes can occur.

Probability Mass Function (PMF) of the Binomial Distribution

The PMF of a binomial distribution, b(k; n, p) = (n choose k) p^k (1-p)^(n-k), specifies the probability of obtaining exactly k successes in n trials. It is essential for calculating the likelihood of outcomes in experiments that have a binary result, such as success or failure.

Monotonicity of Binomial PMF

This concept involves comparing the probabilities of successive outcomes in the binomial distribution. By examining the ratio of b(k; n, p) to b(k-1; n, p), one determines whether the probabilities are increasing or decreasing. This ratio shows that the PMF increases while the number of successes is less than a certain threshold and then decreases after

Mode (Central Term) of the Binomial Distribution

The mode of the binomial distribution is defined as the value of k for which the probability mass function attains its maximum. It is also referred to as the most probable number of successes. This concept is linked to the observation that there is exactly one or sometimes two central terms (if they are equal) where the PMF reaches its highest value.

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