18.X.3. Properties of Binomial Distribution

The expected value E(X) is calculated as:

𝐸(𝑋) = 𝑛𝑝

The mean or expected value of a binomial distribution, denoted as E(X), gives us the average number of successes we expect after conducting a fixed number of independent trials. It’s calculated by multiplying the number of trials (n) by the probability of success (p). For example, if you flip a coin 10 times (n = 10) and the probability of getting heads (success) is 0.5 (p = 0.5), the expected number of heads would be 10 * 0.5 = 5.

Think of a factory that produces light bulbs where each bulb has a 90% chance (0.9) of being good. If the factory tests 100 bulbs, you’d expect about 90 good bulbs on average (100 * 0.9). This helps the factory predict how many bulbs they might have to replace.

The variance of the binomial distribution is given by:

𝑉𝑎𝑟(𝑋) = 𝑛𝑝(1−𝑝)

Variance measures how much the number of successes can vary from the expected number of successes. It considers the probability of success (p) and failure (1-p). If you have a higher variance, the number of successes is more spread out from the mean. For instance, if n = 10 and p = 0.5, the variance would be 10 * 0.5 * (1 - 0.5) = 2.5. A low variance indicates that the outcomes are close to the mean.

Imagine you're rolling a die. If you expect a 3 (which can happen in a predictable way), the variance helps you understand how often you might roll a 1 or a 6 compared to rolling a 3. It gives insight into how likely extreme outcomes are versus outcomes close to the average.

The standard deviation is derived from the variance:

𝜎 = √𝑛𝑝(1−𝑝)

Standard deviation is the square root of variance and provides insight into how much individual outcomes deviate from the mean. By taking the square root, we express this in the same units as the outcomes, making it easier to interpret. If the variance is 2.5, the standard deviation would be √2.5 ≈ 1.58. This means that, on average, the number of successes can vary about 1.58 from the expected number.

Consider a basketball player who scores an average of 20 points per game. Standard deviation reveals how many points they typically score above or below that average. If the standard deviation is 5, you might expect them to score between 15 and 25 points in most games.

Skewness, which measures the asymmetry of the distribution:

𝛾 = (1 − 2𝑝) / √(𝑛𝑝(1−𝑝))

Skewness indicates whether the distribution of successes is symmetrical or tilted toward one side. A skewness of zero implies symmetry, a negative value suggests more successes than failures (left-skewed), and a positive value indicates more failures (right-skewed). It is computed using the values of n and p. For example, if p = 0.7, indicating higher chances of success, the distribution will likely be left-skewed.

Picture a deck of cards. In a game, if winning is easier than losing, more players might end up with wins, leading to a skew. This means that most players successfully win, creating a left-skewed result, as fewer players face failure.

The excess kurtosis indicates the 'tailedness' of the distribution:

𝛾 = (1 − 6𝑝𝑞) / (2𝑛𝑝𝑞)

Kurtosis refers to the presence of outliers in a distribution. A higher kurtosis indicates that the data has heavier tails (more outliers), while a lower kurtosis suggests lighter tails. Excess kurtosis is specifically concerned with how much more extreme the distribution is compared to a normal distribution. If p = 0.5 in a scenario where n is large, the kurtosis will indicate how concentrated the successes are around the mean.

Imagine a thrill-seeking amusement park ride. If most riders rate it either a 1 or a 10 (very low or very high), the ride has high kurtosis because it generates extreme ratings, while a ride that tends to receive a variety of moderate scores would have low kurtosis.

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