Final answer:
The student's question is about finding the medians and modes for a binomial random variable X with given numbers of trials and success probability, and comparing them with the mean. The mean for binomial distributions is found by multiplying the number of trials with the probability of success. The mode is typically the floor of the product (n+1)p, and the median is the value for which the cumulative distribution reaches 0.5.
Step-by-step explanation:
The subject of the question is binomial distribution, which is an important concept in probability theory and statistics. Specifically, the question deals with finding the medians, modes, and comparing them to the mean for a binomial random variable X given certain numbers of trials (n) and success probability (p).
Part A: n = 5, p = 1/3
For a binomial distribution with n = 5 trials and success probability p = 1/3, we identify the median(s), mode(s), and compare them to the mean. The mean (μ) of X is given by μ = np. Here, the mean is 5*(1/3) = 5/3 or approximately 1.67.
The median of a binomial distribution is the middle value(s) when the probabilities are listed in increasing order. It is the value(s) of X for which the cumulative probability function of X reaches 0.5.
The mode of a binomial distribution is the value of X with the highest probability. It is typically the floor of (n+1)p, but when (n+1)p is an integer, there can be two modes: (n+1)p and (n+1)p - 1.
Part B: n = 6, p = 1/3
For n = 6 trials and a success probability of p = 1/3, similar steps as above would be followed to find the median(s), mode(s), and mean. The mean is 6*(1/3) = 2.
In both scenarios, the relation between the median, mode, and mean can be investigated. Typically, for binomial distributions skewed to the right (where 0 < p < 0.5), the mode is less than the median, which is less than the mean. For left-skewed distributions (where 0.5 < p < 1), the reverse is true.