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To find the variance of a hyper geometric random variable in Equation 2.5-2, we used the fact that
Prove this result by making the change of variables
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Probability And Statistical Inference (10th Edition)
- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.arrow_forwardProb. 3 Let X be a random variable with cumulative distribution function (cdf) given by (1-e-x², x ≥ 0 ={1,- x<0 Find the probability that the random variable X falls within one standard deviation of its mean. Fx (x) =arrow_forward2. A random sample X₁, X2, X3 of size 3 is drawn from a population with mean and variance o2. Let 1 3 Đi (Xi+X+X3) - (X₁ + X2 + X3), (3X₁ — X₂ +2X3). - 4 (a) Show that ₁ and ₂ are both unbiased estimators of the population 2 = mean . (b) Compute the variance of ₁ and 2. (c) Which estimator would you use? Explain your answer.arrow_forward
- Suppose that X is a random variable for which the m.g.f. is as follows: M(t) = 1/4(3et+e-t), -∞<t<∞ Find the mean and variance of X.arrow_forwardAnswer no. 1 onlyarrow_forward6) Consider the random variable Zn = cos (2 * T : * Yn/n) where Yn is binomial(n,p) a. What is its asymptotic variance for arbitrary p as n increases b. What happens at p=1/2arrow_forward
- 5arrow_forwardLet X = (X1, X, )" be a bivariate random variable with variance-covariance matrix 4 -1.5 E(X – E(X))(X – E(X))") = ( -1.5 1 You are given that X1 + aX2 is independent of X1. Find the number a. Give your answer in 2 decimal places. Answer:arrow_forwardIf Y is an exponential random variable with parameters beta then mean = E(Y) =beta and variance squaared = V(Y) = beta squared. Show proof of thisarrow_forward
- A discrete random variable X can only take the values -1, 0 and 1. The probabilities of this are P(X = -1) - p, P(X = 1) = p and therefore P(X = 0) = 1 - 2p. Here p is an unknown parameter that we want to estimate. We take a random sample X1, X2, .., Xn and consider two different estimators T1 and T2 for p: #(X; = 1) #(\X;| = 1) %3D en T, = n 2n Here # counts the number of elements, so T2 is the number of random variables that resulted in 1 or -1, divided by 2n. • Calculate the expected mean squared error (MSE) of T2 if p=0.3 and n=200. Give an exact answer. (Correct answer: 3/10000)arrow_forwardExample 5.4 Consider a random sample (X1. X2..X) from an exponential dis- tribution with parameter 2. Consider Ho : A = 2o against H : à > ho Determine the likelihood ratio test associated with the test of Ho against H1.arrow_forwardB) Let X1,X2, .,Xn be a random sample from a N(u, o2) population with both parameters unknown. Consider the two estimators S2 and ô? for o? where S2 is the sample variance, i.e. s2 =E,(X, – X)² and ở² = 'E".,(X1 – X)². [X = =E-, X, is the sample mean]. %3D n-1 Li%3D1 [Hint: a2 (п-1)52 -~x~-1 which has mean (n-1) and variance 2(n-1)] i) Show that S2 is unbiased for o2. Find variance of S2. ii) Find the bias of 62 and the variance of ô2. iii) Show that Mean Square Error (MSE) of ô2 is smaller than MSE of S?. iv) Show that both S2 and ô? are consistent estimators for o?.arrow_forward
- Glencoe Algebra 1, Student Edition, 9780079039897...AlgebraISBN:9780079039897Author:CarterPublisher:McGraw HillTrigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage Learning