3. If X is an exponential random variable with parameter λ > 0, show that for c > 0 cX is exponential with parameter λ/c.
3. If X is an exponential random variable with parameter λ > 0, show that for...
Let X be an exponential random variable with parameter λ, so fX(x) = λe −λxu(x). Find the probability mass function of the the random variable Y = 1, if X < 1/λ Y = 0, if X >= 1/λ
3. Let X be an exponential random variable with parameter 1 = $ > 0, (s is a constant) and let y be an exponential random variable with parameter 1 = X. (a) Give the conditional probability density function of Y given X = x. (b) Determine ElYX]. (c) Find the probability density function of Y.
Problem The random variable X is exponential with parameter 1. Given the value r of X, the random variable Y is exponential with parameter equal to r (and mean 1/r) Note: Some useful integrals, for λ > 0: ar (a) Find the joint PDF of X and Y (b) Find the marginal PDF of Y (c) Find the conditional PDF of X, given that Y 2. (d) Find the conditional expectation of X, given that Y 2 (e) Find the...
5. The Exponential(A) distribution has density f(x) = for x<0' where λ > 0 (a) Show/of(x) dr-1. (b) Find F(x). Of course there is a separate answer for x 2 0 and x <0 (c Let X have an exponential density with parameter λ > 0 Prove the 'Inemoryless" property: P(X > t + s|X > s) = P(X > t) for t > 0 and s > 0. For example, the probability that the conversation lasts at least t...
X is exponential random variable with λ = 3. A) Calculate E(X"2) of this random variable. B) Calculate V(X 2)
Let X be an exponential random variable with parameter A > 0, and let Y be a discrete random variable that takes the values 1 and -1 according to the result of a toss of a fair coin Compute the CDF and the PDF of Z = XY Let X be an exponential random variable with parameter A > 0, and let Y be a discrete random variable that takes the values 1 and -1 according to the result of...
3. Consider a continuous random variable X with pdf given by 0, otherwise This is called the exponential distribution with parameter X. (a) Sketch the pdf and show that this is a true pdf by verifying that it integrates to 1 (b) Find P(X < 1) for λ (c) Find P(X > 1.7) for λ : 1
Simulate n values of an exponential random variable X with parameter λ (of your choice), and compute the sample mean i, sample median m, sample standard deviation s. Plot these quantities as functions of n (on three separate plots). Do x, m, and s converge to any limit values, as n-oo? What are those values and how are they related? Estimate the variance of both x and m for a particular value of n, such as n 100 (by generating,...
Exercise 3. (QUANTIZATION, FROM TEXTBoOK, PROBLEM 4.61) Let X be an exponential random variable with parameter λ. (a) For some d 〉 0 and k a nonnegative integer, find P(kd 〈 X 〈 (k + 1)d) (b) Segment the positive real line into 4 equally probable disjoint intervals.
X is a Poisson random variable of parameter 3 and Y an exponential random variable of parameter 3. Suppose X and Y are independent. Then A Var(2X + 9Y + 1) = 22 B Var(2X + 9Y + 1) = 7 CE[2X2 + 9Y2] = 19 D E[2X2 + 9Y2] = 26