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- Consider two independent exponential random variables X1 and X2 with parameter lambda=1. LetY1 = X1 Y2 = X1 + X2. Find the MMSE estimate of Y1 using Y2.Suppose that X and Y are independent random variables with EX=EY=100, and E[min(X,Y))=79. Find ElX-Y|, (the mean of the absolute value of X-Y).Let X, Y be independent random variables with exponential distribution of parameter θ > 0. Are the random variables Z = X + Y and W = X / (X+Y) independent?
- Let X be a random variable and a real number. Show that E(X - a)² = varX + (µ − a)² Hereμ = EX is the expected value of the random variable X and varX = E(X - μ)^2 is the variance of the random variable X. Guidance: start from the representation - (X-a)^2 = (X µ + μ- a)^2 and group the right side of the representation appropriately into the form (Z + b)^2, where Z is some random variable and b is a real number and open the square. The task should be solved with the help of the expected value calculation rules.Each front tire on a particular type of vehicle is supposed to be filled to a pressure of 26 psi. Suppose the actual air pressure in each tire is a random variable-X for the right tire and Y for the left tire, with joint pdf K(x² + 19 25). (Round your answer to three decimal places.) P(Y > 25) = (c) If the pressure in the right tire is found to be 22 psi, what is the expected pressure in the left tire, and what is the standard deviation of pressure in this tire? 3 (It is known that K = Round your answers to two decimal places.) 350,600 expected pressure psi standard deviation psiSuppose that Y is a uniform continuous random variable on the interval (1,11). Calculate the expected value of the random variable (Y− 1)². Use one decimal place accuracy.
- Let X be a continuous random variable with PDF 3 x > 1 x4 fx(x) = otherwise Find the mean and variance of x.If the random variable X follows the uniform distribution U= (0,1) What is the distribution of the random variable Y= -2lnX. Show its limits.Calculate E{X}, E{Y }, Var{X}, Var{Y } and ρ XY for random variables X and Y with jointdensity function
- Each front tire on a particular type of vehicle pdf f(x, y) = K(x² + y²) 0 supposed to be filled to a pressure of 25 psi. Suppose the actual air pressure in each tire is a random variable, X for the right tire and Y for the left tire, with joint 20≤x≤ 30, 20 ≤ y ≤ 30 otherwise (a) What is the value of K? (Enter your answer as a fraction.) K = (b) What is the probability that both tires are underfilled? (Round your answer to four decimal places.) (c) What is the probability that the difference in air pressure between the two tires is at most 2 psi? (Round your answer to four decimal places.) (d) Determine the (marginal) distribution of air pressure in the right tire alone. for 20 << 30 (e) Are X and Y independent rv's? OYes, f(x, y) = fx(x) fy(y), so X and Y are independent. OYes, f(x, y) + fx(x) fy(y), so X and Y are independent. ONO, f(x, y) = fx(x) fy(y), so X and Y are not independent. ONO, f(x, y) + fx(x) fy(y), so X and Y are not independent.Let X and Y be two random variables with Var(X) = 6, Var(Y ) = 3, andthe correlation ρ(X, Y ) = −1/4. Find the value of Var(X − 2Y + 7).[Joint PDFS will be covered in Week 7] Let the random variables X and Y be the portions of the time in a day that two alternative routes between Topkapi and Uskudar have congestion (X for Route 1 and Y for Route 2). The joint PDF is given by fxy(x.y) = 2x2 + y? where Osxys1. (b) Assume Z=X+Y and W= XY and traffic experts are interested in the expected values of Z and W. E(Z) = and E(W) =O (Simplify your answers. Do not convert fractions into decimals.) (c) Find the variances of X and Y as well as the covariance between them. v(X) =D VY) =D and Cov(X, Y) =D (Simplify your answers. Do not convert fractions into decimals.) (d) The variance of Z can be found as V(Z) =- (Simplify your answer. Do not convert fractions into decimals.)