9)
f(x) = e^(-x)
F(x) = 1 - e^(-x)
P(Y < y)
= P(log x < y)
= P(x < e^y)
= F(e^y)
= 1 - e^(-e^y)
pdf = d/dy F(y)
9. Let X have an exponential distribution with A 1 (see Question 5), and let Y...
In question 5, f(x) = λ*exp(-λx), for x greater or
equal to 0, and zero otherwise.
9. Let X have an exponential distribution with λ = 1 (see Question 5), and let Y = log(X). Find the probability density function of Y. Where is the density non-zero? Note that in this course, log refers to the log base e, or natural log, often symbolized In. The distribution of Y is called the (standard) Gumbel, or extreme value distribution. 2
Let X be distributed as standard exponential
distribution.
a. Let W = αXβ. Find the probability density function and the
cumulative distribution function of W.
b. Let Y = log(W ). Find the probability density function and the
cumulative distribution function of Y .
Question 8. Let X be the Exponential distribution with parameter 2. Let Y=A7. a) Find the distribution function of Y. b) Find the density function of Y. c) Find the distribution of Y.
Let X be an exponential random variable with parameter 1 = 2, and let Y be the random variable defined by Y = 8ex. Compute the distribution function, probability density function, expectation, and variance of Y
Problem 5. Let X be a continuous random variable with a 2-paameter exponential distribution with parameters α = 0.4 and xo = 0.45, ie, ;x 2 0.45 x 〈 0.45 f(x) = (2.5e-2.5 (-0.45) Variable Y is a function of X: a) Find the first order approximation for the expected value and variance of Y b) Find the probability density function (PDF) of Y. c) Find the expected value and variance of Y from its PDF
Problem 5. Let X...
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...
Let X and Y be independent random variables which are exponential with parameter lambda= 1, so then each has probability density function equal to f(x) = exp(-x) when x > 0, and zero otherwise. Compute the probability density function of X + Y . Show detailed explanations and reasoning for each step.
Let random variables X and Y have the bi-variate exponential CDF (cumulative distribution function) : F(x,y) = 1 - exp(-x) - exp(-y) + exp(-x-y-xy) Given x > 0, y>0 a) Determine the probability that 4 < X given that Y = 2 b) Determine the probability that 4 < X given that Y is less than or equal to 2
6. Let X, Y be independent random variables, each having Exponential(A) distribution. What is the conditional density function of X given that Z =
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.