Let X₁, , Xn be a random sample from a large population following a uniform distribution over [0, K]. We do not know K and we want to estimate it. Define - 1/(X₁) ·(X₁ +... + Xn), n (a) What is the expected value of X¡ (for any i = 1,...,n)? (State it as a function of K) (b) Find E(X). (c) Is X an unbiased estimator of K? =

In statistics, the range is computed as the difference between the minimum and maximum values Range(x) = Max(x) - Min(x) If the operatinal time of KAL CULAS graphics cards are each exponentially distribitued with a mean time of 32 months, compute the expected range of i.i.d. random variables X1, X2, X3, X4; where each random variable represents one graphics card to show your work set up all integrals correctly.

'Time headway' in traffic flow is the elapsed time between the time that one car finishes passing a fixed point and the instant that the next car begins to pass that point. Let X be the time headway for two randomly chosen consecutive cars (in seconds). Suppose X has pdf f(x) = {* F(x) (a) Find the value of k. (b) Obtain the mean value of headway and the standard deviation of headway. (c) The cdf of X is = x > 1 x ≤ 1 0 1- x ≤ 1 x>1 Using the cdf, determine the following probabilities. (i) What is the probability that observed depth is at most 3? (ii) What is the probability that observed depth is between 3 and 2?

Let X1,...,Xn be a random sample from the distribution f(x) = 2x, 0 < x < 1. Find the distribution of the sample maximum X(n)

Find a sufficient statistic (in the rigorous sense) for 2 when X Pois(A)

The Volatility X for the S&P stock index on a given day is a normal random variable with mean = 10 and standard deviation = 2 The volatilities recorded over a 100-day period on the S&P500 are Y1, Y2, ... Y100. Assume that these Yi's are independent and identically distributed, uniform on the interval [5,15]. Let V = (Y1 + Y2 + ... + Y100)/100. What approximately is P[9.5 < V < 10.5]?

Suppose X = time to wait for a cab and you know you're going to wait at least 6 minutes, but no more than 10 minutes. Suppose X has a uniform distribution. What is f(x)? f(x) = 1/6 f(x) = 0 f(x) = 1/10 O f(x) = 1/4

X is distributed as a Normal random variable with mean of 100 and a standard deviation of 10 (i.e X~N(100,10). You take a random sample of 20 items from X and you calculate the average of this sample. Of course, someone else could take a random sample of 20 items, calculate the average, and get a slightly different number. The sample average, then, is itself a random variable with its own mean (of 100) and standard deviation. What is the standard deviation of the sample averages?

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 F(x, y) = { K (x² + y²) 20 ≤ ≤ 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≤x≤ 30 (e) Are X and Y independent rv's? Oves, f(x, y) = fx(z) - fy (y), so X and Y are independent. Oves, f(x, y) + fx(z) - fy(y), so X and Y are independent. ONO, f(x, y) = fx(2) - fy(y), so X and Y are not independent. ONO, f(z,y) #fx(z) - fy(y), so X and Y are not independent.

X is distributed as a Normal random variable with mean of 100 and a standard deviation of 10 (i.e X~N(100,10). You take a random sample of 80 items from X and you calculate the average of this sample. Of course, someone else could take a random sample of 80 items, calculate the average, and get a slightly different number. The sample average, then, is itself a random variable with its own mean (of 100) and standard deviation. What is the standard deviation of the sample averages? (Please report your answer to two decimal places, such as 5.67.)