4. Suppose that X and Y are independent and follow an exponential distri- bution with parameter...
Let X1,... , Xn be independent random variables, each following an exponential distri- bution with rate λ. Let Y = min(X1, .. . , Xn). Find the cd.f. and pdf. of Y. HINT:
6. Let X and Y be two independent samples of a standard uniform distri- bution. Let Z be the closest integer to X/Y (i.e. the value that we get by rounding X/Y). Is Z more likely to be even or odd? (hint: draw the sample space over X and Y and identify the regions where Z is even or odd. Work out when Z = 0, when Z-1, and so forth)
4. Suppose that X is a random variable having the following probability distri- bution function - 0 if r<1 1/2 if 1 x <3 1 if z 2 6 (a) Find the probability mass function of X. (b) Find the mathematical expectation and the variance of X (c) Find P(4 X < 6) and P(1 < X < 6). (d) Find E(3x -6X2) and Var(3X-4).
Problem 4 (Conditional Expectation and Variance). Suppose the joint distri- bution of (X, Y) is given by the following contingency (row represents x) table 20 points (x,y) 2 4 6 1 0.3 0 0.1 2 0 0.2 0 3 0.1 0 0.3 A) Compute the marginal distributions of X and Y B) Are X and Y independent? Explain. C) Find the conditional distribution of Y given X -1 D) Compute E[Y|X 1] E) Compute EY|X= 2] F Compute E[exp(X)Y|x 2
Example 3.6. Take a random sample of size n from an exponential distri- bution with rate parameter XA. 1. Derive an exact 95% confidence interval for X. 2. Suppose your sample is of size 9 and has sample mean 3.93. (a) What is your 95% confidence interval for λ? (b) What is your 95% confidence interval for the population mean? 3. Repeat the above using the CLT approximation (rather than an eract interval
rate parameter A, for y independent rate parameter A, for X,. Let Y be the minimum of all these n random variables, i.e., Y- min(X1, X2,... ,Xn). Show that Y is distributed as exponential with rate Problem 6. Let X1, X2,..., Xn be independent exponential random variables with rn.
Problem 1 (16 points). Suppose that X and Y are independent random variables and that Y follows a geometric distribution with parameter p. Assume that X takes only nonnegative integer values, and let Gx(z) be the probability generating function of X. (We make no additional assumptions about the distribution of X.) Show that P(X<Y) = Gx(1- p). Clearly indicate the step(s) in your argument that use the assumption that X and Y are independent.
Suppose that random variables X and Y are independent. Further, X is an exponential random variable with parameter 1 = 3, and Y is an uniformly distributed random variable on the interval (0,4). Find the correlation between X and Y, rounded to nearest .xx
Suppose X has an exponential distribution with parameter λ = 1 and Y |X = x has a Poisson distribution with parameter x. Generate at least 1000 random samples from the marginal distribution of Y and make a probability histogram.
(1 point) In Unit 3, I claimed that the sum of independent, identically distributed exponential random variables is a gamma random variable. Now that we know about moment generating functions, we can prove it. Let X be exponential with mean A 4. The density is 4 a) Find the moment generating function of X, and evaluate at t 3.9 The mgf of a gamma is more tedious to find, so l'll give it to you here. Let W Gamma(n, A...