Question

If \(x=r\) is the unique mode of a distribution that is \(b(n, p)\), show that $$ (n+1) p-11\).

Short answer

The mode 'r' of a binomial distribution \(b(n, p)\) lies within the range \((n+1) p-1 < r < (n+1) p\). This is determined by finding the values of 'x' where the ratio of the probability of 'x+1' successes to 'x' successes is greater than or equal to 1.

Step-by-step solution

Understanding the binomial probability function

The binomial probability function \(p(x)\) for 'x' successes in 'n' trials is defined as \(p(x) = C(n, x) \cdot p^x \cdot (1-p)^{n-x}\), where \(C(n, x)\) is the number of combinations of 'n' items taken 'x' at a time. It represents the probability of getting 'x' successes in 'n' trials.

Formulating the ratio \(p(x+1) / p(x)\)

The hint suggests calculating the ratio \(p(x+1) / p(x)\), i.e., probability of 'x+1' successes to probability of 'x' successes. This comes out to \(\frac{C(n, x+1) \cdot p^{x+1} \cdot (1-p)^{n-(x+1)}}{C(n, x) \cdot p^x \cdot (1-p)^{n-x}}\). We can simplify this to \(\frac{(n - x) \cdot p}{(x+1) \cdot (1 - p)}\).

Finding range within which ratio is greater than 1

Now, we need to find values of 'x' where the ratio \(p(x+1) / p(x)\) is greater than 1. That means \(\frac{(n - x) \cdot p}{(x+1) \cdot (1 - p)} > 1\) . Solving this inequality simplifies to \(x < (n + 1) p - 1\). Similarly, to be the mode, the probability p(x) should be equal or greater than p(x+1), which gives the inequality \(x > (n + 1) p - 1\). Consequently, the mode 'r' lies between \((n+1) p-1\) and \((n+1) p\).

Key concepts

Understanding the Binomial Probability Function

The binomial probability function is a cornerstone concept in statistics that describes the likelihood of a specified number of successes in a series of independent trials. For instance, tossing a coin 'n' times and wanting to know the probability of it landing heads 'x' times can be calculated using this function. The function is mathematically expressed as:

\[ p(x) = C(n, x) \cdot p^x \cdot (1-p)^{n-x} \]

where:

• \( C(n, x) \) represents the combinatorial number of ways to choose 'x' successes from 'n' trials.
• \( p \) is the probability of success on a single trial.
• \( 1-p \) is the probability of failure on a single trial.

Understanding this function helps us calculate the probability of events where there are two outcomes like success/failure, pass/fail, or win/lose, which is fundamental in many fields such as finance, genetics, and quality control.

Deciphering Probability Inequalities

Probability inequalities are essential in understanding how probabilities of different events compare. In the context of the binomial distribution, these inequalities help us figure out when an event becomes more likely than not. When we compute the ratio
\( p(x+1) / p(x) > 1 \)
, we’re looking for when the probability of having one more success (\( x+1 \)) exceeds the probability of the current number of successes (\( x \)).

The ratio itself simplifies to
\[ \frac{(n - x) \cdot p}{(x+1) \cdot (1 - p)} \]
, which can be used to establish crucial thresholds. For example, in the case of the binomial distribution’s mode, we seek for the range within which this ratio is greater than 1. Solving the inequality tells us that if you have a certain number of successes, adding one more makes the event less likely, as you've exceeded the mode.

Discrete Random Variables and Binomial Distribution

Discrete random variables come into play when we are dealing with events that can be counted, like the number of heads in coin tosses. In a binomial distribution, our discrete random variable 'x' represents the count of successes in 'n' trials, where the trials are independent, and each has the same probability of success 'p'. Understanding discrete random variables is crucial because they allow us to compute various probabilities and to make predictions about the outcomes of certain processes.

One of the most important characteristics of a discrete random variable is its mode — the value that appears most frequently. Knowing how to determine the mode of a binomial distribution helps us in identifying the most probable number of successes, which can be a vital insight in planning and decision-making processes. The problem solved above illustrates how mathematical concepts like function ratios and inequalities are applied to discover the mode of a discrete random variable within a binomial setting.