Final answer:
The random variables in question are binomial random variables if they adhere to the criteria of a binomial experiment with independent trials, two possible outcomes, and constant success probability. The variable X would represent the number of posted questions to the listserv on a randomly picked day, ranging from 0 to the total possible number of questions. The probability for a given range is found by summing the individual probabilities calculated using the binomial formula.
Step-by-step explanation:
To determine whether the random variables in question are binomial random variables, we must first revisit the definition of a binomial distribution. A binomial random variable arises from a set of independent Bernoulli trials with two possible outcomes, success or failure, and the probability of success (p) and failure (q) remains constant throughout the trials. Given these conditions of independence, a fixed number of trials, consistency of success probability, and dichotomous outcomes, we can say that a random variable is binomially distributed.
Based on the information provided, if the experiment in question adheres to these criteria, then:
- The random variable X would be defined as the number of posted questions to the listserv on a randomly picked day.
- The values that X may take on could range from 0 to the maximum number of questions that could be posted in a day (presumably the number of trials).
- The distribution of X, assuming it is binomial, would be given as X~B(n, p), where 'n' is the total number of independent trials (or opportunities for posting a question) and 'p' is the probability of a question being posted in one trial. The mean and standard deviation would be calculated using the formulas μ=np and σ=√npq respectively.
- To find the probability of 10 to 14 questions being posted, we would sum the individual probabilities P(X = x) for x ranging from 10 to 14, using the appropriate binomial probability formula.
Without specific values for 'n' and 'p', we cannot provide a numerical answer for part D. However, the process would involve calculating the probability for each value of X and summing them up.