21.12. Suppose that X and Y are independent, nonnegative random variables and that E[X]=∞ and E[Y]=0. What is the value common to E[XY] and E[X]E[Y]? Use the conventions (15.2) for both the product of the random variables and the product of their expected values. What if E[X] =∞ and 0 < E[Y]<∞o?
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21.12. Suppose that X and Y are independent, nonnegative random variables and
and E[Y]=0. What is the value common to E[XY] and
E[X]E[Y]? Use the conventions (15.2) for both the product of the random
variables and the product of their expected values. What if E[X] =∞ and
0<E[Y]<∞o?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe3d557df-911d-4c1c-951f-1faffc0aa610%2F3b69b262-bc63-4c28-b5c8-8fa113649d42%2Fmfmbpi6_processed.jpeg&w=3840&q=75)
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- 5. Suppose that X1, X2, .. , Xn are an iid sample from the distribution given below, f(X)e = (1/2)(1+ xe); –1 < X < 1. Show that 3X is a consistent estimator of e4. Suppose X₁ and X₂ are independent random variables with cdf Fx(x) = sin(x), 0 ≤ x ≤ 1/ a. Show that fx(2) (x) = 2sin(x) cos(x), where X (2) is the random variable for the 2nd order statistic. (Remember, we only have 2 random variables.) b. Show that fx(1)(x) = 2cos(r) [1 sin(x)], where X(1) is the random variable for the 1st order statistic.1. Suppose we have a random sample X₁, X2, ..., Xn from an Exponential distribution with mean 1/A. Suppose we want to estimate the mean 1/A. One estimator for 1/X is T₁ = X. Of interest is to note that the minimum of X₁, X2,..., Xn, say X(1), has an Exponential distribution with mean (n)-¹. (a) Show that T₁ is an unbiased estimator of 1/A. (b) Find the constant a such that T₂ = aX(1) is an unbiased estimator of 1/X. (c) Since T₁ and T₂ are both unbiased we prefer the estimator with smaller variance. Which of the estimators T₁ and T₂ would you choose for estimating the mean 1/X?
- 1. The time to failure of an AC compressor is a random variable T, with the following pdf. 200 f(t) = (10 + t)3 (a) Find R(t). (b) Determine the reliability for the first year of operation. (c) Find the probability of failure occurring in the interval of time [15,20]. (d) Find the design life if a reliability of 0.99 is desired.1. Let X be a Poisson random variable on the non-negative integers with rate λ = 4. Let W = 2X + 10. (a) What is the range of W? (b) Find a formula for Pw(k).Let X1,... Xn be independent random variables, all having the same distribution with expected value u and variance o?. The random variable X, defined as the arithmetic average of these variables, is called the sample mean. That is, the sample mean is given by (a) Show that E[X] = µ. (b) Show that Var[X] = o²/n. The random variable S2, defined by EL (Xi – X)² п — 1 is the sample variance. (Denominator is n – 1, not n, due to (d).) (c) Show that (Xi – X)? = E-, X? – nX.
- 1. Find the population covariance between X and Y .2. Find E[Y |X = x].3. Find Var[Y |X = x].7. a) Suppose that X is a uniform continuous random variable where 02. A random variable X has a p.d.f. f(x) given by √(1-x)6 02 f(x₁y) = {√ x² + 20₁0 < x < 1,0 <1 <2, 0 elsewhere 1 g(x) = 2x(x + 3/3), h(y) = 1/32 + 1/ / a. Determine the expected value of x b. Determine the expected value of Y C. Determine the expected value of 2=X+YSuppose that X1, X2, ..., Xn is a random sample with a probability law P(x) = {0x, 0 Where 8 is an unknown parameter. si x = 1,2,3, 4; otherwise A. Get an estimator of 0 by the method of the moments. B. Prove that the estimator is unbiased.4.Suppose that X is a random variable for which E(X)=µ and Var(X)=o². Show that E(X(X-1)]=µ(µ-1)+o?SEE MORE QUESTIONSRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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