a. Suppose X is a random variable with mean μ. What can you say about P(X> 3u)? b. Suppose in addition, that X has variance o². What can you say about P(u - 30 < X < µ +30)?
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- 2x + 1 25 a. P(X= 4) b. P(X -10) 3.1.11 WP SS VS f(x) = x = 0, 1, 2, 3, 4 3.3.3 WP VS Determine the mean and variance of the random variable in Exercise 3.1.11.Let X1, X2, X3, and X4 be a random sample of observations from a population with mean μ and variance σ2. Consider the following estimator of μ: �"!= 0.15 X1 + 0.35 X2 + 0.20 X3 + 0.30 X4.What is the variance of � "!Let X1, X2, .. ., Xn be a random sample from a normal distribution with mean u and variance o?. Find an estimator for o? using the method of moments. Is this estimator bi- ased/unbiased? Is it consistent?
- Say that we have a null hypothesis that X1,..., X16 that are Normally distributed with mean 0 and variance 1. Call the sample mean X. Consider the test where p-value = 20 (4 X) - 24 (4|X) – 1 We will reject this if p-value < a. a) Show that the type I error rate of this test is a. b) Say that X1,..., X16 that are Normally distributed with mean 1 and variance 1. What is the type II error rate of this test when a = 0.05? •..Assume that you have a sample of n, = 8, with the sample mean X, = 47, and a sample standard deviation of S, = 7, and you have an independent sample of n, = 7 from another population with a sample mean of X, = 37 and the sample standard deviation S, = 8. Assuming the population variances are equal, at the 0.01 level of significance, is there evidence that u, > H2? Determine the hypotheses. Choose the correct answer below. O A. Ho: H1 SH2 H1: Hy> H2 O B. Ho: H1 =H2 H: H1 # H2 O D. Ho: H1 H2 O C. Ho: H1> H2 H1: H1 SH2When we consider a random sample of size n as X,..., X, which is from Binomial (n, 6). X+1 a. Is a biased estimator for the parameter 8? Is this estimator asymptotically unbiased? n+1 b. Find the mean squared error (MSE) of the estimator. Investigate its MSE consistency.
- Let Y1 and Y2 be independent random variables with unknown mean μ and known variance σ2 =1. Which of the following two estimators of μ has smaller MSE? Justify your answer.Assume X and Y are independent random variables with X ~ Binomial(n =100, p = 0.2) and Y ~ Pascal(m =16, p = 2/3) . Find the indicated expected value and variance. a. E(3X + 2Y +1) b. Var(3X + 2Y +1)8.3
- 1. Six batteries taken at random from a large lot were subjected to life tests with the following results: 19, 22, 18, 16, 25, and 20 hours. Find the probability that the sample mean of x= 20 hours, as observed here, or one less than 20 hours could arise if the true mean of the lot of batteries is 21.5 hours, assuming a known variance of o = 10 hours, for the individual batteries in the lot. %3DLet X-1: correct, X=0 : wrong And P= Pr(X=1)=2/5 What is the variance? .32 O.6 O .4just please answer the sub-b