7 Let X be a Bernoulli random variable with P(X = 1) = 4. Consider a...
7. Let X1,....Xn random sample from a Bernoulli distribution with parameter p. A random variable X with Bernoulli distribution has a probability mass function (pmf) of with E(X) = p and Var(X) = p(1-p). (a) Find the method of moments (MOM) estimator of p. (b) Find a sufficient statistic for p. (Hint: Be careful when you write the joint pmf. Don't forget to sum the whole power of each term, that is, for the second term you will have (1...
Problem 4: Consider the problem of estimating the unknown parameter p of a Bernoulli random variable that describes the probability that a coin toss results in a head. Denote by X the outcome of the jth toss of the coin and let j-1 denote the sample mean. Part I: Use Chebyshev inequality to determine the number of tosses n needed so that P( -pl> 0.01) 0.01 The estimate should be independent of p Part II: Compute ElIX -pl]. Your answer...
The Bernoulli random variable takes values 0 and 1, and has probability function f(x) = px (1 − p)1−x(a) By calculating f(0) and f(1), give a practical example of a Bernoulli experiment, and a Bernoulli random variable. (b) Calculate the mean and variance of the Bernoulli random variable.
2. Let X be a Bernoulli random variable with probability of X -1 being a. a) Write down the probability mass function p(X) of X in terms of a. Mark the range of a (b) Find the mean value mx(a) EX] of X, as a function of a (c) Find the variance σ剤a) IX-mx)2) of X, as a function of a. (d) Consider another random variable Y as a function of X: Y = g(X) =-log p(X) where the binary...
4(25 points) Let X be a random variable with mean μ = E(X) and σ2 V(X). Let X = n Σ_1Xī be X2 + Xs) be the average of the the sample mean from a random sample (X X. Let X (X first three observations. (a) Prove that X is an unbiased estimator for μ. Prove that X is also an unbiased estimator for μ. (b) Explain that X is a consistent estimator for μ. Explain why X is not...
Problem:2 Let Xi, X2,... , Xn be a random sample from Bernoulli(p) and consider es- timators iand p2- i. Compute the mean square error (MSEp) for both estimators p? and p2 Note that you must show the details of the calculation to receive full credit. ii. Use R to plot MSE, for both estimators using sample sizes n 20 and n- 300. Comment on the plots. iii. Use R to simulate 10,000 different Bernoulli samples of n 300 with success...
Dr. Beldi Qiang STATWOB Flotllework #1 1. Let X.,No X~ be a i.İ.d sample form Exp(1), and Y-Σ-x. (a) Use CLT to get a large sample distribution of Y (b) For n 100, give an approximation for P(Y> 100) (c) Let X be the sample mean, then approximate P(.IX <1.2) for n 100. x, from CDF F(r)-1-1/z for 1 e li,00) and ,ero 2Consider a random sample Xi.x, 、 otherwise. (a) Find the limiting distribution of Xim the smallest order...
Let X1 Xn be a random sample of size n from a Bernoulli population with parameter p. Show that p= X is the UMVUE for p. 5.4.22 Let X1 Xn be a random sample of size n from a Bernoulli population with parameter p. Show that p= X is the UMVUE for p. 5.4.22
(4) (20 pts) Let X ~ Bernoulli(p). Define Z = 3x-1. (a) Is Z a random variable? Why or why not? (b) Show that EIZ 2p. (c) Show that EZ2 4p. (d) What is the variance of Z?
Problem 5 Let Xi, X2, ..., Xn be a random sample from Bernoulli(p), 0 < p < 1, and 7.i. Prove that the sample proportion is an unbiased estimator of p, i.e. p,- is an unbiased estimator of p 7.ii. Derive an expression for the variance of p,n 7.iii. Prove that the sample proportion is a consistent estimator of p. 7.iv. Prove that pn(1- Pn)