Recall that the binomial distribution with parameters n and p is governed by Let n be some known number, say n-1000. Th...
Consider the binomial distribution with parameters n 10 and p (unknown) a) Is this binomial distribution an exponential family distribution? b) Find a sufficient statistic for p.
Recall from the slides that the Gamma distribution can be reparameterized using the two parameters The pdf looks like the shape parameter, and μ the mean. 02 Let 6-( (^ ) and rewrite this as the pdf of a 2-parameter exponential family. Enter η (0)-T (y) below. θ = Recall from the slides that the Gamma distribution can be reparameterized using the two parameters The pdf looks like the shape parameter, and μ the mean. 02 Let 6-( (^ )...
Let M have a binomial distribution with parameters N and p. Conditioned on M, the random variable X has a binomial distribution with parameters M and (a) Determine the marginal distribution for X (b) Determine the covariance between X and Y M- X
Let X be a random variable, which has a binomial distribution with parameters n and p. It is known that E(X) = 12 and Var(X) = 4. Find n and p.
Please answer as neatly as possible. Much thanks in advance! 7. If Y has a binomial distribution with parameters n and p, consider p = (Y + 1)/(n+2) as an estimator of p. Is p a consistent estimator for p? Is p an asymptotically unbiased estimator for p? 7. If Y has a binomial distribution with parameters n and p, consider p = (Y + 1)/(n+2) as an estimator of p. Is p a consistent estimator for p? Is p...
Exercise 1: Let Y be a binomial distribution of parameters n = 10 and p = 0.4. Determine the normal approximation for: a) P (Y ≥ 7) Exercise 2: The quality control of an automobile manufacturer accuses 1% of failures in the process of antioxidant protection of bodywork of the vehicles it produces. Calculate the probability that: a) none of the 100 vehicles ordered by a dealership has the aforementioned failure; b) only one vehicle has the fault mentioned in...
(c) Idifficult] Let x ~ Binomial (n, p) where n is an even number. Find the PMF of Y g(X) mod (X, 2) where "mod" denotes modulus division of the first argument by the second argument (d) Idifficult MA] Let X~NegBin (k, p). Find the PMF of Y g(X) = mod(X, n) where n E N
(2) Let Y be a binomial random variable with parameters n and p. Remember that E(Y) V(Y)p1 -p) We know that Y/n is an unbiased estimator of p. Now we want to estimate the variance of Y with n(2(1 (a) Find the expected value of this estimator (b) Find an unbiased estimator that is a simple modification of the proposed estimator
Exercise 7 (2 points) Recall the binomial coefficient for integer parameters 0 Sk< n. Prove that Exercise 8 (2 points) Prove the following: if z is an integer with at most three decimal digits aia2a3, then x is divisible by 3 if and only if aut a2 +a3 is divisible by 3. Exercise 9 (3 points) A square number is an integer that is the square of another integer. Let x and y be two integers, each of which can...
he second form for one-parameter exponential family distributions, introduced during lecture 09.1, was Jy (y | θ) = b(y)ec(0)t(y)-d(0) Let η = c(0). If c is an invertible function, we can rewrite (1) as where η is called the natural, or canonical, parameter and K(n) = d(C-1(n)). Expression (2) is referred to as the canonical representation of the exponential family distribution (a) Function κ(η) is called the log-normalizer: it ensures that the distribution fy(y n) integrates to one. Show that,...