Question 3: Bernoulli distribution (23/100 points) Consider a random sample X1,...,Xn from a Bernoulli distribution with unknown parameter p that describes the probability that Xi is equal to 1. That is, Bernoulli(p), i = 1, ..., n. (10) The maximum likelihood (ML) estimator for p is given by ÔML = x (11) n It holds that NPML BIN(n,p). (12) 3.a) (1 point) Give the conservative 100(1 – a)% two-sided equal-tailed confidence interval for p based on ÔML for a given...
(b). Let x, X.,...,x. be a random sample from the Bernoulli (0) distribution 0). Find the most powerful test 1,:0= versus H,:6-of size a=0,011. (). What is the power of this test?
For each Bernoulli process, find the expected number of successes: 1. Number of trials =10, Probability of success =0.6 2. Number of trials =210, Probability of success =1/10. 3. Number of trials =43, Probability of success =0.3. 4. Number of trials =23, Probability of failure =0.8. 5. Number of trials =59, Probability of failure =2/7.
Let Sı, S2,... , Sn be a random sample from a Bernoulli distribution with pmf (a) Calculate E(S) and find the MOM estimator of p. (b) Construct the log-likelihood function and use this to find the max- imum likeligood (ML) estimator of p.
Let Sı, S2,... , Sn be a random sample from a Bernoulli distribution with pmf (a) Calculate E(S) and find the MOM estimator of p. (b) Construct the log-likelihood function and use this to find the max-...
m 1: Suppose that.X, form a random sample from a Bernoulli distribution for s unknown (0 θ < 1). Suppose also that the prior distribu- the beta distribution with parameters a >0 and 8> 0. Then the posterior distribution which the value of the parameter i of θ given that Xi z, (i l, where -n isthe beta distribution with parameters (0.3.), -: Proof:
m 1: Suppose that.X, form a random sample from a Bernoulli distribution for s unknown (0...
4. Let X1, X2, ..., Xn be iid from the Bernoulli distribution with common probability mass function Px(x) = p*(1 – p)1-x for x = 0,1, and 0 < p < 1 14 a. (4) Find the MLE Ôule of p.
A geometric distribution is a series of independent Bernoulli trials. True or false?
1. Suppose Yi,½, , Yn is an iid sample from a Bernoulli(p) population distribution, where 0< p<1 is unknown. The population pmf is py(ulp) otherwise 0, (a) Prove that Y is the maximum likelihood estimator of p. (b) Find the maximum likelihood estimator of T(p)-loglp/(1 - p)], the log-odds of p.
1. Suppose Yi,½, , Yn is an iid sample from a Bernoulli(p) population distribution, where 0
a) Consider the following data on a variable that has Bernoulli distribution: X P (X) 0 0.3 1 0.7 Find the Expected value and the variance of X. And E(X)-X Px) b) Consider the following information for a binomial distribution: N number of trials or experiments 5 x- number of success 3 Probability of success p 0.4 and probability of failure 1-p 0.6 Find the probability of 3 successes out of 5 trials: Note P(x) Nox p* (1-p)Note: NcN!x! (N-x)!...
Exercise 2. Let Xn, n EN, be a Bernoulli process uith parameter p = 1/2. Define N = min(n > 1:X,メ } For any n 2 1, define Yn = XN4n-2. Show that P(Yn = 1) = 1/2, but Yn, n E N is not a Bernoulli process
Exercise 2. Let Xn, n EN, be a Bernoulli process uith parameter p = 1/2. Define N = min(n > 1:X,メ } For any n 2 1, define Yn = XN4n-2. Show...