Question

In: Statistics and Probability

Suppose that X is a single observation from a Binomial(n, p) distribution where n is known...

Suppose that X is a single observation from a Binomial(n, p) distribution where n is known and 0 < p < 1 is unknown. Consider three estimators of p:

pˆ = X /n “sample proportion”

pˆA = (X + 2)/ (n + 4) “plus four estimator”

pˆB = (X + (√n/4))/( n + √ n ). “constant MSE estimator”

(a) Find the bias functions for all three estimators.

(b) Find the variance functions of all three estimators.

(c) Find the MSE functions of all three estimators.

(d) Create plots of the bias, variance, and MSE of all three estimators when n = 4. Note: I encourage you to use software of your choice. If you plot by hand, make sure that you have well defined scales and that you can see where the functions cross one another. That is, your plots should be more than just rough sketches. (e) Repeat the previous part for n = 40.

(f) Repeat the previous part for n = 400.

(g) Using your plots as evidence, discuss the relative merits of the three estimators. What are some advantages/disadvantages of each?

Solutions

Expert Solution

orchestra answered 2 years ago
Next > < Previous

Related Solutions

Suppose that X is a single observation from a Binomial(n, p) distribution where n is known...
Suppose that X is a single observation from a Binomial(n, p) distribution where n is known and 0 < p < 1 is unknown. Consider three estimators of p: pˆ = X /n “sample proportion” pˆA = (X + 2)/ (n + 4) “plus four estimator” pˆB = (X + (√n/4))/( n + √ n ). “constant MSE estimator” (a) Find the bias functions for all three estimators. (b) Find the variance functions of all three estimators. (c) Find the...
Binomial Distribution. Suppose that X has a binomial distribution with n = 50 and p =...
Binomial Distribution. Suppose that X has a binomial distribution with n = 50 and p = 0.6. Use Minitab to simulate 40 values of X. MTB > random 40 c1; SUBC > binomial 50 0.6. Note: To find P(X < k) for any k > 0, use ‘cdf’ command; this works by typing: MTB > cdf; SUBC > binomial 50 0.6. (a) What proportion of your values are less than 30? (b) What is the exact probability that X will...
Suppose that X ~ NB(r, p) [negative binomial distribution] and that Y ~ B(n, p) [binomial]....
Suppose that X ~ NB(r, p) [negative binomial distribution] and that Y ~ B(n, p) [binomial]. a. Give a probabilistic argument to show that P(X > n) = P(Y < r). b. Use the FPF to express the equality in part (a) in terms of PMFs. c. Using the complementation rule, how many terms of the PMF of X must be evaluated to determine P(X > n)? d. How many terms of the PMF of Y must be evaluated to...
Suppose that x has a binomial distribution with n = 201 and p = 0.49. (Round...
Suppose that x has a binomial distribution with n = 201 and p = 0.49. (Round np and n(1-p) answers to 2 decimal places. Round your answers to 4 decimal places. Round z values to 2 decimal places. Round the intermediate value (σ) to 4 decimal places.) (a) Show that the normal approximation to the binomial can appropriately be used to calculate probabilities about x. np n(1 – p) Both np and n(1 – p) ≤ or ≥ 5 (b)...
Suppose that x has a binomial distribution with n = 50 and p = 0.6, so...
Suppose that x has a binomial distribution with n = 50 and p = 0.6, so that μ = np = 30 and σ = np(1 − p) = 3.4641. Approximate the following probabilities using the normal approximation with the continuity correction. (Hint: 25 < x < 39 is the same as 26 ≤ x ≤ 38. Round your answers to four decimal places.) (a) P(x = 30) (b) P(x = 25) ( c) P(x ≤ 25) (d) P(25 ≤...
For the binomial distribution with n = 10 and p = 0.4, where p is probability...
For the binomial distribution with n = 10 and p = 0.4, where p is probability of success. Let X be the number of successes. (a) Find the probability of three or more successes. (b) Find the µ, E(X), and σ 2 , V ar(X)
Suppose a random variable, x, arises from a binomial experiment. Suppose n = 6, and p...
Suppose a random variable, x, arises from a binomial experiment. Suppose n = 6, and p = 0.80. Find: a.) P(x = 0) b.) P(x = 1) c.) P(x = 2) d.) P(x = 3) e.) P(x = 4) f.) P(x = 5) g.) P(x = 6) h.) the sum of probabilities calculated in parts (a) through (g) i.) the population mean ? for this probability distribution j.) the population standard deviation ? for this probability distribution Enter answers rounded...
For a binomial probability distribution, n = 130 and p = 0.60. Let x be the...
For a binomial probability distribution, n = 130 and p = 0.60. Let x be the number of successes in 130 trials. a. Find the mean and standard deviation of this binomial distribution. a. Find the mean and the standard deviation of this binomial distribution. b. Find to 4 decimal places P(x ≤ 75) using the normal approximation. P(x ≤ 75) = c. Find to 4 decimal places P(67 ≤ x ≤ 72) using the normal approximation. P(67 ≤ x...
1. Suppose X has a binomial distribution with parameters n and p. Then its moment-generating function...
1. Suppose X has a binomial distribution with parameters n and p. Then its moment-generating function is M(t) = (1 − p + pe^t ) n . (a) Use the m.g.f. to show that E(X) = np and Var(X) = np(1 − p). (b) Prove that the formula for the m.g.f. given above is correct. Hint: the binomial theorem says that Xn x=0 n x a^x b^(n−x) = (a + b)^n . 2. Suppose X has a Poisson distribution with...
Suppose a random variable, x, arises from a binomial experiment. If n = 23, and p=...
Suppose a random variable, x, arises from a binomial experiment. If n = 23, and p= 0.22, find the following probabilities using technology. show work P (x = 21) P (x = 6) P (x = 12) P (x<=14) P (x >=17) 6. P (x <= 9)
ADVERTISEMENT
Subjects
ADVERTISEMENT
Latest Questions
ADVERTISEMENT