Question

The mean of the Bincmial distribution with \(n\) observations and probability of success \(p\), is (a) \(p q\) (b) \(n p\) (c) \(\sqrt{n p}\) (d) \(\sqrt{p q}\).

Short answer

(b) \(np\)

Step-by-step solution

Understanding the Binomial Distribution

A Binomial Distribution arises when an experiment is repeated a fixed number of times \(n\), each with two possible outcomes: success (with probability \(p\)) and failure (with probability \(q = 1 - p\)).

Recall the Formula for the Mean

In a Binomial Distribution, the mean (also called the expected value) is given by \(np\), where \(n\) is the number of trials and \(p\) is the probability of success on each trial.

Identify the Correct Answer

Given the formula for the mean \(np\), the correct answer among the options provided is (b) \(np\).

Key concepts

Mean of Binomial Distribution

The mean of a Binomial Distribution is an important concept as it helps summarize the central tendency of the distribution. In statistics, the mean is often referred to as the expected value. For the Binomial Distribution, the mean provides insight into the average number of successful outcomes you would expect, given a specific number of trials and a constant probability of success. The formula for calculating the mean of a Binomial Distribution is: \[ \mu = np \] where \(n\) is the total number of trials and \(p\) is the probability of a successful outcome in each trial. For example, if you flip a coin 100 times (where \(n = 100\)), and the probability of flipping heads each time is 0.5 (\(p = 0.5\)), the mean number of heads you would expect is \(np = 100 \times 0.5 = 50\). This formula allows you to predict and understand the average outcome over many repetitions of the same experiment.

Expected Value

The term "expected value" is used interchangeably with the mean in the context of probability distributions. It represents the average value you expect to obtain from an experiment over a large number of trials. In the Binomial Distribution, the expected value quantifies the center of the distribution, providing a measure of central tendency, similar to the mean in regular statistics. For the Binomial Distribution, the expected value is calculated using the formula: \[ E(X) = np \] Here, \(E(X)\) signifies the expected value of the random variable \(X\), which in the case of a Binomial Distribution is the number of successful trials. Understanding the expected value helps interpret real-world scenarios in a probabilistic sense, giving you a clear idea of what to anticipate over repeated processes.

Probability of Success

In Binomial Distribution, the probability of success is a key parameter. It reflects the chance of obtaining a successful outcome in one trial of a repeated experiment. Denoted as \(p\), it is essential to the formulation of the Binomial Distribution. Consider an experiment like flipping a coin. Here, you may define success as getting heads, in which case \(p = 0.5\) since there is a 50% chance of getting heads. The probability of failure \(q\) is complementary to \(p\), calculated as \(q = 1 - p\). It's crucial to ensure that the probability of success remains constant across the trials for the situation to be modeled accurately as a Binomial Distribution. This consistency is what characterizes the trials as independent and identically distributed, foundational in using the Binomial model.

Binomial Distribution Formula

The formula for a Binomial Distribution is fundamental in calculating probabilities for the number of successful outcomes in a fixed number of identical experiments. This formula considers both the number of trials \(n\) and the probability of success \(p\). The general probability distribution function for a Binomial Distribution is: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where

• \(P(X = k)\) is the probability of \(k\) successes in \(n\) trials,
• \(\binom{n}{k}\) is the binomial coefficient or "n choose k," calculated as \(\frac{n!}{k!(n-k)!}\),
• \(p^k\) is the probability of success raised to the power of the number of successes,
• \((1-p)^{n-k}\) is the probability of failure to the power of the number of failures.

This formula allows you to ascertain the likelihood of observing exactly \(k\) successes over \(n\) trials, given the parameters \(n\) and \(p\). It's crucial for tasks in statistics that require modeling binary outcomes efficiently and precisely.