Suppose that X is a Poisson random variable with Round your answers to 3 decimal places (e.g. 0.987). (a) Compute the exact probability that X is less than 14. With continuity correction: i = (b) Use normal approximation to approximate the probability that X is less than 14. Without continuity correction: i 21. With continuity correction: i (c) Use normal approximation to approximate the probability that 28 < X < 42. Without continuity correction: i

When a man observed a sobriety checkpoint conducted by a police​ department, he saw 656 drivers were screened and 5 were arrested for driving while intoxicated. Based on those​ results, we can estimate that P(W)equals=0.00762​,where W denotes the event of screening a driver and getting someone who is intoxicated. What does P(W) ​denote, and what is its​ value? What does P(W) ​represent? P(W)=

Let X ~ Geom(p), a geometric distribution with parameter p € (0, 1). This is a discrete probability distribution on the positive integers, with p.m.f. given by fx (k) = (1 − p)k-¹p, k = = 1, 2,... (i) Verify (by calculation) that the log-likelihood function for the parameter p is given by l(x;p) = n (logp + (x − 1) log(1 − p)). - (ii) Derive the MLE estimator for p, and use it to compute the MLE estimate for the following sample data ¹: 1 3 1 1 1 2 3 1 1 1 1 (iii) What is the MLE estimator for 1/p? Verify this estimator is unbiased.

Let X equal the IQ of a randomly selected American. Assume X ~ N( μ μ {"version":"1.1","math":"μ"} =100, σ σ {"version":"1.1","math":"σ"} =4). What is the probability that a randomly selected American has an IQ below 90?

Suppose you repeatedly flip a fair coin. Use the normal approximation discussed above to (approximately) compute the probability that it takes more than 210 flips to get 100 heads (you should use the continuity correction). The coin is unfair with probability of observing a head being 1/3. Suppose you repeatedly flip this unfair coin. Use the normal approximation discussed above to (approximately) compute the probability that it takes more than 210 flips to get 60 heads (you should again use the continuity correction).

Compute​ P(X) using the binomial probability formula. Then determine whether the normal distribution can be used to estimate this probability. If​ so, approximate​ P(X) using the normal distribution and compare the result with the exact probability. n=56  p=0.5   x=30  A)For n=56​, p=0.5​, and X=30, use the binomial probability formula to find​ P(X).     B)Can the normal distribution be used to approximate this​ probability? A. ​No, because np(1−p)≤10   B. ​Yes, because np(1−p)≥10   C. ​Yes, because np(1−p)≥10   D. ​No, because np(1−p)≤10   C) Approximate​ P(X) using the normal distribution. Use a standard normal distribution table. A. ​P(X)=enter your response here ​(Round to four decimal places as​ needed.)   B. The normal distribution cannot be used.     D) By how much do the exact and approximated probabilities​ differ? A.enter your response here ​(Round to four decimal places as​ needed.) B. The normal distribution cannot be used.   E)By…

Show that the expectation of the exponential distribution with param- eter A is 1/A.