1. Let the random variable Y have pdf as f(y\A) = e®=e=t(e-1), y>0,1>0. Show that W = e – 1] ~ x² or, equivalently, U = e – 1~x.
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![1. Let the random variable Y have pdf as
f(y)A) = e"=e{(e*-1), y>0, A > 0.
Show that W = {e* – 1] ~ xỉ or, equivalently, U = e' –1~}X.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2e8b9c67-63da-4856-8557-a4b1bba4357d%2F725f19ff-af92-4d7f-b6d7-f0820878d15b%2Fmyubvao_processed.jpeg&w=3840&q=75)
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- 3. The length of time required by students to complete a 1 hour exam is a random variable with a pdf given by: f (x) = = ca + for 0 sæ<1 a. Find c. Enter c as a reduced fraction. C = b. Find F(x). Enter coefficients as reduced fractions and use ^ to denote powers. F(x) = c. Find the probability that a student takes less than 30 minutes to complete the exam. Enter the probability as a reduced fraction. prob = d. Find the median length of time to take the exam. Enter your answer in hours with 4 decimal places. median = hours e. Find the length of time for the first 10% of the students to complete the exam. Enter your answer in hours with 4 decimal places. hours answer = f. Find the expected value, variance, and standard deviation of X. Enter your answer in hours with 4 decimal places (or hours squared in the case of variance), hours expected value =Show that if β^1 is conditionally unbiased, then it is unbiased; that is, showthat if E(β^1|X1, . . . . , Xn) = β1, then E(β^1) = β1.Show that variance o? = (x²) – ((x))²
- Suppose that {Yt} is (weakly) stationary with autocovariance function Yk. (a) Find an expression for yk in terms of yo = of correlation) Var(Yt). (Think about the definitionA study reports data on the effects of the drug tamoxifen on change in the level of cortisol-binding globulin (CBG) of patients during treatment. With age = x and ACBG = y, summary values are n = 26, Ex; = 1618, (x,x)² = 3756.96, Ey, = 281.9, (y;-)²: = 465.34, and Ixy = = 16,718. (a) Compute a 90% CI for the true correlation coefficient p. (Round your answers to four decimal places.) (b) Test Ho: P = -0.5 versus H₂: P < -0.5 at level 0.05. Calculate the test statistic and determine the P-value. (Round your test statistic to two decimal places and your P-value to four decimal places.) z = P-value = State the conclusion in the problem context. O Fail to reject Ho. There is evidence that p < -0.5. Fail to reject Ho. There is no evidence that p < -0.5. O Reject Ho. There is no evidence that p < -0.5. Reject Ho. There is evidence that p < -0.5. (c) In a regression analysis of y on x, what proportion of variation in change of cortisol-binding globulin level could be explained by variation in…Q5 Find the variance for the PDF px(x) = e-«/2, x > 0.
- let x be a random variable with moment generating function Mx(t)=(0.6 + 0.4e^t)^20 then the variance of x isThe pdf is given as follows: f(x) = { (1/6)e^(1/6x) ; x > 0 0; ew] (i) Find the mean and variance of X using the pdf. (ii) Find the moment generating function of X. (iii) Find the mean and variance of X using the moment generating function.If(x,,y; ).(x,, y; ),..(x,,y«) be a random sample taken from a logistic regression, then the log-likelihood cquation is, a) ,In| Eln(1-z,) -T, b) Σh(1 -π) Σy n -Σh(π) 1-7, c) i-1 1-1 d) Σyn(1- r)+Σ -
- Show that H {(12)) is not normal in S3. However, why is H' = ((123)) normal in S3?differentiate H with respect to G H-3e20G Select one: O a. H' = 60e 20G - 3/5G^(4/3) Ob. H' = 60e^20G + 4/3G^(4/3) O c. none of the choices Od. H' = 60e 20G-4/3G^(4/3) Oe. H' = 20e^20G-4/3G^(4/3)Let (xi, Yi) (i = 1,...,n) be independent following the Poisson regression model (without intercept) Y~ Poission (x₁) (This is often used to model number of events occurred given a covariate. For example, the number of traffic accidents in a month as Y; at the i-th intersection vs the number of cars going through the same intersection in the same period, as x₂) (a) Since E(Y;) = Ari, we can write Y₁ = λr; + e; where E(e;) = 0. Find the least square estimator for A. (b) Write the likelihood function L(A) (treat x₁,...,En as non-random) (c) Find MLE for A. (d) Suppose we use Gamma(a, ß) with pdf Ba π(X) = -Aa-le-BA for A>0 r(a) as the prior distribution. Find the posterior distribution given observations (y₁, ₁),..., (Yn, In). And find the corresponding Bayes estimator for A. [Hint: The prior is a conjugate prior so the posterior distribution is also a Gamma distribution. The mean of the Gamma(a, b) distribution is a/b]