Question

In: Statistics and Probability

Let X be an exponential distribution with mean=1, i.e. f(x)=e^-x for 0<X< ∞, and 0 elsewhere....

Let X be an exponential distribution with mean=1, i.e. f(x)=e^-x for 0<X< ∞, and 0 elsewhere. Find the density function and cdf of

a) X^1/2

b)X=e^x

c)X=1/X

Which of the random variables-X, X^1/2, e^x, 1/X does not have a finite mean?

Solutions

Expert Solution


Related Solutions

Let f(x) = a(e-2x – e-6x), for x ≥ 0, and f(x)=0 elsewhere. a) Find a...
Let f(x) = a(e-2x – e-6x), for x ≥ 0, and f(x)=0 elsewhere. a) Find a so that f(x) is a probability density function b)What is P(X<=1)
Consider an exponential distribution f(x|θ) = θe^(−θx) for x > 0. Let the prior distribution for...
Consider an exponential distribution f(x|θ) = θe^(−θx) for x > 0. Let the prior distribution for θ be f(θ) = e^ −θ for θ > 0. (a) Show that the posterior distribution is a Gamma distribution. With what parameters? (b) Find the Bayes’ estimator for θ.
Let f(x) = {(C/x^n if 1≤ x <∞; 0 elsewhere)} where n is an integer >1....
Let f(x) = {(C/x^n if 1≤ x <∞; 0 elsewhere)} where n is an integer >1. a. Find the value of the constant C (in terms of n) that makes this a probability density function. b. For what values of n does the expected value E(X) exist? Why? c. For what values of n does the variance var(X) exist? Why?
Let X be a exponential random variable with pdf f(x) = λe−λx for x > 0,...
Let X be a exponential random variable with pdf f(x) = λe−λx for x > 0, and cumulative distribution function F(x). (a) Show that F(x) = 1−e −λx for x > 0, and show that this function satisfies the requirements of a cdf (state what these are, and show that they are met). [4 marks] (b) Draw f(x) and F(x) in separate graphs. Define, and identify F(x) in the graph of f(x), and vice versa. [Hint: write the mathematical relationships,...
Let A = 0 1 1 0 (a) Calculate the matrix exponential e^(At). (Hint: It might...
Let A = 0 1 1 0 (a) Calculate the matrix exponential e^(At). (Hint: It might help to write down the power series expansions for the hyperbolic functions cosh(t) =(e^t + e^(−t))/2 and sinh(t) =(e^t −e^(−t))/2 and then try to write eAt in terms of these two functions.) (b) Use your matrix from part (a) to solve the nonhomogeneous initial value problem x' = 0 1 1 0 x + 2 -1 , x(0) = 1 2 . (Hint: You...
1. Let X1, X2 be i.i.d with this distribution: f(x) = 3e cx, x ≥ 0...
1. Let X1, X2 be i.i.d with this distribution: f(x) = 3e cx, x ≥ 0 a. Find the value of c b. Recognize this as a famous distribution that we’ve learned in class. Using your knowledge of this distribution, find the t such that P(X1 > t) = 0.98. c. Let M = max(X1, X2). Find P(M < 10)
Let f(x, y) be a function such that f(0, 0) = 1, f(0, 1) = 2,...
Let f(x, y) be a function such that f(0, 0) = 1, f(0, 1) = 2, f(1, 0) = 3, f(1, 1) = 5, f(2, 0) = 5, f(2, 1) = 10. Determine the Lagrange interpolation F(x, y) that interpolates the above data. Use Lagrangian bi-variate interpolation to solve this and also show the working steps.
Let X have a Poisson distribution with mean μ . Find E(X(X-1)) and use it to...
Let X have a Poisson distribution with mean μ . Find E(X(X-1)) and use it to prove that μ = σ 2 .
The exponential distribution (?)E(λ) with density ?(?)=??−??f(x)=λe−λx for all ?>0x>0. Suppose that ?=2λ=2. Find the median...
The exponential distribution (?)E(λ) with density ?(?)=??−??f(x)=λe−λx for all ?>0x>0. Suppose that ?=2λ=2. Find the median of ?X. Find the expected value and variance of ?X. Find P(?>3)
Let X be a random variable with CDF F(x) = e-e(µ-x)/β, where β > 0 and...
Let X be a random variable with CDF F(x) = e-e(µ-x)/β, where β > 0 and -∞ < µ, x < ∞. 1. What is the median of X? 2. Obtain the PDF of X. Use R to plot, in the range -10<x<30, the pdf for µ = 2, β = 5. 3. Draw a random sample of size 1000 from f(x) for µ = 2, β = 5 and draw a histogram of the values in the random sample...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT