Final answer:
X represents the sum of 250 iid random draws and follows a binomial distribution with parameters n = 250 and p = 0.90. The mean is 225 and the standard deviation is approximately 4.74.
Step-by-step explanation:
If we consider X as the sum of 250 iid(random independent and identically distributed) random draws from the population Y, where P(Y=1)=0.90 and P(Y=0)=0.10, we need to determine which distribution best describes X. The variable Y takes on two values, 0 and 1, with given probabilities, so each draw from Y can be considered a Bernoulli trial. Since X is the sum of 250 Bernoulli trials, it will follow a binomial distribution with parameters n = 250 and p = 0.90. Therefore, the correct choice is:
b) X follows a binomial distribution.
By the Central Limit Theorem, we know that for a large number of draws (n is sufficiently large), the binomial distribution begins to approximate a normal distribution. However, the sum of the draws inherently describes a binomial distribution, not the normal distribution. It would not be a uniform distribution since the outcomes are not equally likely, and it would not be an exponential distribution since that describes the time between events in a Poisson process, not the sum of a series of Bernoulli trials.
The mean (μ) of a binomial distribution is given by μ = np, which in this case would be μ = 250 * 0.90 = 225. The standard deviation (σ) is given by σ = √(np(1-p)), which calculates to approximately σ = √(250 * 0.90 * 0.10) ≈ 4.74.