Question

Let X∼Bin(n,p)

(a) For n=5,p=1/3, find all medians and all modes of X. How do they compare to the mean?

Answer 1

To find the medians and modes of a binomial distribution with n = 5 and p = 1/3, we can use the binomial probability formula. The medians are the values of X that have a cumulative probability of 0.5, and the modes are the values of X with the highest binomial probability. Comparing the medians and modes to the mean can help us understand their relationship.

Step-by-step explanation:

The binomial distribution is used to model the probability of a certain number of successes, X, in a fixed number of independent trials, n, each with the same probability of success, p.

For the given values of n = 5 and p = 1/3, we can find the medians and modes of X.

Medians:

Since X is a discrete random variable, the median(s) of X can be found by finding the value(s) of X that have a cumulative probability of 0.5. In other words, we need to find the value(s) of X for which P(X ≤ x) = 0.5 and P(X ≥ x) = 0.5.

For n = 5 and p = 1/3, the possible values of X are 0, 1, 2, 3, 4, 5.

1. To find the lower median, we look for the smallest value of X for which P(X ≤ x) = 0.5.
2. To find the upper median, we look for the largest value of X for which P(X ≥ x) = 0.5.

Using the binomial probability formula P(X = x) = (nCx) * (p^x) * (q^(n-x)), we can calculate the cumulative probabilities for each value of X and find the medians.

Modes:

The mode(s) of X are the value(s) of X that occur with the highest probability. In a binomial distribution, the mode(s) can be found by looking for the value(s) of X with the highest binomial probability.

Using the binomial probability formula, we can calculate the probability for each value of X and find the mode(s).

Comparison to the Mean:

The mean of a binomial distribution is given by μ = np. In this case, the mean is μ = 5 * 1/3 = 5/3. We can compare the medians and modes to the mean to see if they are similar or different.