Note:Due to insufficient time i couldn't solved remaining bits of c.If you want to answer for remaining bits of c please post it separately.Thank you.
LetX,X2, , XnLLd. Bernoulli(p), and let Y-Σ,Xi. Then we know that Y-Binomial(n, p). 5. Consider t...
4. Let X,X,Bernoulli(p), and let Y Xi. Then we know that Y-Binomial(n.p) Consider the hypotheses Hop-Po against Hip#po- a. c. For the particular case of po0.25 and n 5, fill in the table: 0 3 4 λ(y) P(Y - y) Find the (generalized) likelihood ratio test φ(y) of size α for testing Ho:p-po vs. H,: p d. po. Your test should be expressed in terms of y and α.
1(a) Let Xi, X2, the random interval (ay,, b%) around 9, where Y, = max(Xi,X2 ,X), a and b are constants such that 1 S a <b. Find the confidence level of this interval. Xi, X, want to test H0: θ-ya versus H1: θ> %. Suppose we set our decision rule as reject Ho , X, be a random sample from the Uniform (0, θ) distribution. Consider (b) ,X5 is a random sample from the Bernoulli (0) distribution, 0 <...
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
2. i) Let B be a random variable with the Binomial (n, p) distribution, so that Write down the likelihood function L(p) for m independent observations xi,...,Inm 2 marks 6 marks ili) Compute the bias and the mean squared error of the corresponding maximum likeli- from B. Int ii) Show that the maximum likelihood estimate for pis-Σ.ri. mn [7 marks] hood estimator of p. iv) Let X be a continuous random variable with density function for x > 0, and...
Assume that we have three independent observations: where Xi ~ Binomial(n 7,p) for i E { 1.2.3). The value of p E (0, 1) is not known. When we have observations like this from different, independent ran- dom variables, we can find joint probabilities by multiplying together th ndividual probabilities. For example This should remind you the discussion on statistical independence of random variables that can be found in the course book (see page 22) Answer the following questions a...
e (4 marks) Let m be an integer with the property that m 2 2. Consider that X1, X2,.. ., Xm are independent Binomial(n,p) random variables, where n is known and p is unknown. Note that p E (0,1). Write down the expression of the likelihood function We assume that min(x1, . . . ,xm) 〈 n and max(x1, . . . ,xm) 〉 0 5 marks) Find , and give all possible solutions to the equation dL dL -...
problems binomial random, veriable has the moment generating function, y(t)=E eux 1. A nd+ 1-p)n. Show that EIX|-np and Var(X) np(1-p) using that EIX)-v(0) nd E.X2 =ψ (0). 2. Lex X be uniformly distributed over (a b). Show that ElXI 쌓 and Var(X) = (b and second moments of this random variable where the pdf of X is (x)N of a continuous randonn variable is defined as E[X"-广.nf(z)dz. )a using the first Note that the nth moment 3. Show that...