Consider the given probability histogram of a binomial random variable.
Binomial Random Variable ($n=5, p=0.5$)
What are the center and shape of the distribution?
A. Center: 2 Shape: symmetric
B. Center: 2.5 Shape: symmetric
C. Center: 2.5 Shape: slightly skewed
D. Center: 3 Shape: uniform
Answer (1)
C e n t er : 2.5 S ha p e : sy mm e t r i c
Explanation
Understand the problem We are given a binomial random variable with parameters n = 5 and p = 0.5 . We need to find the center and shape of the distribution.
Calculate the center The center of a binomial distribution is given by its mean, which is calculated as μ = n p . In this case, n = 5 and p = 0.5 , so the center is: μ = 5 × 0.5 = 2.5
Determine the shape The shape of a binomial distribution depends on the value of p . When p = 0.5 , the distribution is symmetric. This is because the probabilities of success and failure are equal.
State the answer Therefore, the center of the distribution is 2.5 and the shape is symmetric.
Examples
Understanding the center and shape of a binomial distribution is useful in many real-world scenarios. For example, if you flip a fair coin 5 times, the binomial distribution with n = 5 and p = 0.5 describes the probability of getting a certain number of heads. The center (mean) tells you the average number of heads you'd expect to see over many sets of 5 flips, and the symmetric shape indicates that getting a number of heads above the mean is as likely as getting a number of heads below the mean.
