Mean, Variance, and Standard Deviation In Exercises 11–14, find the mean, variance, and standard deviation of the binomial distribution with the given values of n and p.
n = 50, p = 0.4

Verified step by step guidance

1. Step 1: Recall the formulas for the mean, variance, and standard deviation of a binomial distribution. The mean (μ) is given by μ = n * p, the variance (σ²) is given by σ² = n * p * (1 - p), and the standard deviation (σ) is the square root of the variance, σ = √(σ²).
2. Step 2: Substitute the given values of n = 50 and p = 0.4 into the formula for the mean. Compute μ = 50 * 0.4.
3. Step 3: Substitute the same values of n = 50 and p = 0.4 into the formula for the variance. Compute σ² = 50 * 0.4 * (1 - 0.4).
4. Step 4: Use the result from Step 3 to calculate the standard deviation. Compute σ = √(σ²), where σ² is the variance obtained in the previous step.
5. Step 5: Summarize the results by stating the mean, variance, and standard deviation of the binomial distribution based on the calculations from the previous steps.

Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is characterized by two parameters: n (the number of trials) and p (the probability of success on each trial). This distribution is useful in scenarios where there are two possible outcomes, such as success or failure.

Mean of a Binomial Distribution

The mean of a binomial distribution, also known as the expected value, is calculated using the formula μ = n * p. This value represents the average number of successes expected in n trials. For the given parameters n = 50 and p = 0.4, the mean provides a central value around which the distribution of outcomes is centered.

Variance and Standard Deviation

Variance measures the spread of a distribution and is calculated for a binomial distribution using the formula σ² = n * p * (1 - p). The standard deviation, which is the square root of the variance, indicates how much the outcomes deviate from the mean. These measures are essential for understanding the variability in the number of successes in the given binomial trials.