a) A statistic T is said to be a consistent estimator of parameter if T converges in probability
Bernoulli Law of Large Numbers:
In n trails let X denotes the number of successes with constant p of success for each trial, then for any ,however small
b) Also
Property:
If t is consistent for
Then by the property of consistency is a consistent estimator for p(1-p)
Also is consistent for
For each n, let Xn be a binomial random variable with n trials and probability of...
Let N be a binomial random variable with n = 2 trials and success probability p = 0.5. Let X and Y be uniform random variables on [0, 1] and that X, Y, N are mutually independent. Find the probability density function for Z = NXY.
The random variable X counting the number of successes in n independent trials is a Binomial random variable with probability of success p. The estimator p-hat = X/n. What is the expected value E(p-hat)? Op O V(np(1-p)) Опр O p/n Submit Answer Tries 0/2
Let X1, X2, .., Xn be a random sample from Binomial(1,p) (i.e. n Bernoulli trials). Thus, п Y- ΣΧ i=1 is Binomial (n,p). a. Show that X = ± i is an unbiased estimator of p. Р(1-р) b. Show that Var(X) X(1-X (п —. c. Show that E P(1-р) d. Find the value of c so that cX(1-X) is an unbiased estimator of Var(X): п
Let A > 0 be fixed and for each n - 1,2,3.., let Xn be a Binomial Random variable with parameters n, and pn -^. (i.e The number of trials is n and thıe success probability is pn --) (a) Write the moment-generating-function, Mx (t of X,. (You do not have to 72 derive it from scratch. You may use the general formula for the mgf of a binomial variable as provided in the appendix of the text). (b) Show...
Show that if X follows a binomial distribution with n, trials and probability of success p,-p,jz 1,2, Hint: Use the moment generating function of Bernoulli random variable) 1. , n and X, are independent then X, follows a binomial distribution.
QUESTION 1 Consider a random variable with a binomial distribution, with 35 trials and probability of success equals to 0.5. The expected value of this random variable is equal to: (Use one two decimals in your answer) QUESTION 2 Consider a random variable with a binomial distribution, with 10 trials and probability of success equals to 0.54. The probability of 4 successes in 10 trials is equal to (Use three decimals in your answer) QUESTION 3 Consider a random variable...
5. For each n = 1, 2, . . . , the random variable Xn is such that P(Xn-01 = 1-1 and PlXn-θ + n) n. Show that Xn is biased estimator of θ and find its MSE. Show that Xn is consistent estimator of θ.
12. The random variable Y obeys the binomial distribution with number of trials n and success probability p. (a) Derive the MGF for Y. (b) Use the MGF to find the mean and standard deviation of Y.
Suppose X is a Binomial random variable for which there are 3 independent trials and probability of success 0.5. What is the mean? Suppose Y is a Binomial random variable for which there are 5 independent trials and probability of success 0.5. What is the mean?
For a Binomial random variable, the probability of exactly zero successes out of two trials equals 0.0289. What is the associated probability of "success," p?