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Assume that X~ Bin(n, p) Find the variance of X algebraically. Hint: First find E(X(X-1)) Use...
7. Find the variance of X, given X ∼ Bin(n = 15, p = 0.13) a. 1.3025 b. 13.05 c. 1.6965 d. 1.95
3. Let X~ Bin(n,p) with n known (a) State the parameter space for the mode b) State EX] and V[x]. (c) Is p an unbiased estimator for the population proportion p? Show why or why not (d) To estimate the variance of X, we generally use θ 2Pl1 ow is a estimator for V지. (e) Modify 0 from part (b) to form an unbiased estimator for V[X ].
1. Let X ~ Bin(n = 12, p = 0.4) and Y Bin(n = 12, p = 0.6), and suppose that X and Y are independent. Answer the following True/False questions. (a) E[X] + E[Y] = 12. (b) Var(X) = Var(Y). (c) P(X<3) + P(Y < 8) = 1. (d) P(X < 6) + P(Y < 6) = 1. (e) Cov(X,Y) = 0.
a) Prove algebraically that(m+n | p+n)≥(m | p) for all m, p, n ∈ N and such that m≥p. b) Prove the above inequality by providing a combinatorial proof. Hint: this can be done by creating a story to count the RHS exactly (and explain why that count is correct), and then providing justification as to why the LHS counts a larger number of options. a) Prove algebraically that p for all m, p, n EN, and such that m...
(2 pts) Verify that for j > 1: P(Bin(n, p) = j) * P(Bin(n,p) =– 1) 1 Now use this fact to determine the value of j for which P(Bin(n, p) = j) is highest.
6. Suppose C) ~ N (C), Ģ:: ru). Find the distribution of X|Y. Hint use the formula p p(y) 7. Consider i.i.d. observations Xi, .., Xn ~ N(H, 1) (a) Compute E(XiX). Hint: use the above problem, and find the conditional distribution of Xi given X first (b) Compute E (ix)
Compute f '(a) algebraically for the given value of a. HINT [See Example 1.] f(x) = x2 − 9; a = 1 Compute f '(a) algebraically for the given value of a. HINT [See Example 1.] f(x) = x3 + 9x; a = 5 Compute the derivative function f '(x) algebraically. HINT [See Examples 2 and 3.] f(x) = x2 − 8 Compute the derivative function f '(x) algebraically. HINT [See Examples 2 and 3.] f(x) = 2x − 1 Find the equation...
1. Find Fx in terms of φ (t). Is X a continuous random variable ? 2. Compute p(X 0) 3. Compute E(X). Hint: use the CDF expectation formula, and integration by parts. You may assume that lim, t"o(-t) 0 for all n 2 0. 4. Find the CDF Fx (u) 5. Compute V(X). Hint: use Fxa, and follow the same hint of part (3) 1. Find Fx in terms of φ (t). Is X a continuous random variable ? 2....
Let X be binomially distributed to parameters n and p. Find E?X2? (Hint: write X as a sum of n indicator variables, what is then X2?)
5) Consider the polynomial P() z2-z-1. (a) Find two integers n, m E Z, so that P(x) has a zero in [n, m. (b) Use the bisection method twice to get an approximation to the zero of P(x) in n, m] (c) Use Newton's method twice to get an approximation to the zero of P() in n,m (d) Use the quadratic formula to find the actual zero of P() in [n, m (e) Compute the relative %-error for each of...