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- T has an exponential distribution such that P(T≤2)=2P(T>4) What is Var(T)=?1. Consider the Gaussian distribution N (m, σ2).(a) Show that the pdf integrates to 1.(b) Show that the mean is m and the variance is σ.The moment generating function can be usedto find the mean and variance of the normal distribution.a. Use derivatives of MX(t) to verify that E(X)=meanand V(X) =varianceb. Repeat (a) using RX(t)=ln[MX(t)], and comparewith part (a) in terms of effort.
- Consider the joint PDF f(x,y)=4xy for 0<x<1 and 0<y<1. (a) Are X and Y independent?(b) Find both marginal distributions. (c) Find both conditional distributions.Henry performed a two-tailed test for an experiment in which N = 24. He could not find his table of t critical values, but he remembered the tcv at df= 13. He decided to compare his tobt with this tcv. Is he more likely to make a Type I or a Type II error in this situation?A CI is desired for the true average stray-load loss ? (watts) for a certain type of induction motor when the line current is held at 10 amps for a speed of 1500 rpm. Assume that stray-load loss is normally distributed with ? = 2.2. (Round your answers to two decimal places.) (a) Compute a 95% CI for ? when n = 25 and x = 53.7. , watts(b) Compute a 95% CI for ? when n = 100 and x = 53.7. , watts(c) Compute a 99% CI for ? when n = 100 and x = 53.7. , watts(d) Compute an 82% CI for ? when n = 100 and x = 53.7. , watts(e) How large must n be if the width of the 99% interval for ? is to be 1.0? (Round your answer up to the nearest whole number.)n = You may need to use the appropriate table in the Appendix of Tables to answer this question.
- What is the critical value for a two-tailed t test when α = 0.02 and d.f. = 18?Derive a general expression for the variance of net future loss Var(L) with a constant force of interest δ under each of the following models:i. μ_x(t) = μii. μ_x(t) = 1/(ω-x)Denise runs a dependent samples t-test and calculates r2subjects=.46 and r2condition=.23. a) SScondition is what proportion of SSsubjects (give in 2 decimals)?
- Q4) If X is a continuous random variable having pdf ke~ (2x+3y) x>0y>0 xy) = = e p(x) { 0 otherwise Find a) the constant k b) P(X>1) ¢) X, X2, 02, standard deviation.Given the moment-generating function MX(t) =e3t+8t2 , find the moment-generating function of the ran-dom variable Z = 1 4 (X − 3), and use it to determine the mean and the variance of Z.For a certain insured, the distribution of aggregate claims is binomial with parameters m = 12 and q = 0.25. The insurer will pay a dividend, D, equal to the excess of 80% of the premium over claims (excess = 0.8 x premium - claims), if positive. The premium is 5. Calculate E[D].