2. If X and Y are independent random variables, X has a normal distribution with mean...
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;
Suppose X and Y are independent random variables with Exponential(2) distribution (Section 6.3). We say X ~ Exponential(2) if its pdf is f(x) = -1/2 for x > 0.
10) The X random variable has a normal distribution. P(X > 15) = 0.0082 and P(X<5) = 0.6554 find the mean and variance of this distribution
Let X1, X2,..., X, be n independent random variables sharing the same probability distribution with mean y and variance o? (> 1). Then, as n tends to infinity the distribution of the following random variable X1 + X2 + ... + x, nu vno converges to Select one: A. an exponential distribution B. a normal distribution with parameters hi and o? C a normal distribution with parameters 0 and 1 D. a Poisson distribution
4.3. Let X and Y be independent random variables uniformly distributed over the interval [θ-, θ + ] for some fixed θ. Show that W X-Y has a distribution that is independent of θ with density function for lwl > 1.
Problem 9: 10 points Suppose that X, Y are two independent identically distributed random variables with the density function f(x)= λ exp (-Az), for >0. Consider T- and find its cumulative distribution function and density function.
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.
Problem 8: 10 points Suppose that (X, Y) are two independent identically distributed random variables with the density function defined as f (x) λ exp (-Ar) , for x > 0. For the ratio, z-y, find the cumulative distribution function and density function.
6. Find the variance of the F distribution for v2> 4, by making use of the definition of F in Theo 8.14 of the F distribution for v2 > 4, by making use of the definition of F in Theorem (in Your Textbook) and the fact that for a random variable V having the chi-square distribution with v2 degrees of freedom
4. Let 8 >0. Let X, X2,..., X, be a random sample from the distribution with probability density function S(*;ð) - ma t?e-vor x>0, zero otherwise. Recall: W=vX has Gamma( a -6, 0-ta) distribution. Y=ZVX; = Z W; has a Gamma ( a =6n, = ta) distribution. i=1 E(Xk) - I( 2k+6) 120 ok k>-3. 42 S. A method of moments estimator of 8 is 42.n 8 = h) Suggest a confidence interval for 8 with (1 - 0) 100%...