Let X and Y be independent exponentially distribution
random variables with rate α and β respectively. Find P (X > Y
).
Let X and Y be independent exponentially distribution random variables with rate α and β respectively....
Let X, Y be two independent exponential random variables with means 1 and 3, respectively. Find P(X> Y)
6.81 Let Yı, Y. ..., Y, be independent, exponentially distributed random variables with mean B. a Show that Y) = min(Y , Y2, ..., Y,) has an exponential distribution, with mean B/n. b If n = 5 and B = 2, find P(Ym <3.6).
2. Let X and Y be independent, exponentially distributed random variables where X has mean 1/λ and Y has mean 1/μ. (a) What is the joint p.d.f of X and Y? (b) Set up a double integral for determining Pt <X <Y) (c) Evaluate the above integral. (d) Which of the following equations true, and which are false? {Z > t} = {X > t, Y > t} (e) Compute P[Z> t) wheret 0. (f) Compute the p.d.f. of Z.
3. A random variable X is said to have a Cauchy(α, β) distribution if and only if it has PDF function Now, suppose that Xi and X2 are independent Cauchy(0, 1) random variables, and let Y = X1 + X2. Use the transformation technique to find and identify the distribution of Y by first finding the joint distribution of Xi and Y. (Seahin 3 4
4. Let X, Y, and Z be independent random variables, each with the standard normal distribution. Compute the following: (a) P[X + Y> Z +2 (b) Var3x 4Y;
2. Let X and Y be independent, exponentially distributed random variables where X has mean 1/λ and Y has mean 11. (a) What is the joint p.d.f of X and Y? (b) Set up a double integral for determining Pt < X <Y). (c) Evaluate the above integral. (d) Which of the following equations true, and which are false? (e) Compute PIZ> t where t20. (f) Compute the pd.f. of Z. Z = min(X,Y)
3. Suppose that X and Y are independent exponentially distributed random variables with parameter λ, and further suppose that U is a uniformly distributed random variable between 0 and 1 that is independent from X and Y. Calculate Pr(X<U< Y) and estimate numerically (based on a visual plot, for example) the value of λ that maximizes this probability.
Let Xi and X2 independent random variables, with distribution functions F1, and F2, respectively Let Y a Bernoulli random variable with parameter p. Suppose that Y, X1 and X2 are independent. Proof using the de finition of distribution function that the the distribution function of Z =Y Xit(1-Y)X2 is F = pF14(1-p)F2 Don't use generatinq moment functions, characteristic functions) Xi and X2 independent random variables, with distribution functions F1, and F2, respectively Let Y a Bernoulli random variable with parameter...
(25 points,) Let X and Y be two independent and identically distributed random variables that have exponential distribution with rates 1 respectively. Find the distribution of Note: you can give either cdf or pdf) (25 points,) Let X and Y be two independent and identically distributed random variables that have exponential distribution with rates 1 respectively. Find the distribution of Note: you can give either cdf or pdf)
f a random sample X,X, X, from the 2. Let Y, < Y.< Y, be the order statistics o exponential distribution with mean β. Let (i) Are the random variables U,V,W independent? (ii) What is the distribution of each of U,V and W.