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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- 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 SH2If the random variable X as a mean of u and a variance of o2 what is the P((X-u)² < 20): O less that equal to 1/4 O greater than or equal to 3/4 cannot tell O equal to 5/4Suppose that X₁,..., Xn are i.i.d. from a normal population with mean μ and variance o². Calculate the expected value and the variance of the sample mean X and find the distribution of √n(X − µ).
- X is a normally distributed random variable with a mean of 5 and a standard deviation of 2. For which value of a is P(X > a) = 0.6293? What is a?If X is a negative binomial rv, then Y= r+Xis the total number of trials necessary to obtainr S’s. Obtain the mgf of Y and then its mean valueand variance. Are the mean and variance intuitively consistent with the expressions for E(X)and V(X)? ExplainAssume that a random variable Y has a mean of zero and standard deviation of one. Calculate the expected value of X1 and X2 if they are defined as below. X1 = Sigma+ Mu X2 = Sigma(Y+Mu)
- The random variable Z has mean 30 and variance 225. Calculate the limits between which you would expect the bulk of observations to lie for the random variable Z/3 − 7.1. 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. %3DHow do you find the mean and variance of part(d)?I need the detailed solution.
- 6. Find the median of an exponential random variable X with parameter 1 = 4. (The median of X is m iff P(X < m) = 0.5.) Does it differ from E(X)? Try to explain.(c) Which estimator, Ô, or Ö, is a more efficient estimator of 0? Explain your answer.Suppose that x has a beta distribution with parameters α = 2.5 and β = 1. Determine the following: a. P(X < 0.25) b. P(0.25 < X < 0.75) c. Mean and variance
