It turns out that the "Pocket Change" data set was taken from a random variable: X that was exponentially distributed with mean μ = 0.50 . In other words, the random variable X is the amount of change in pocket
(a) What is the PDF for X ∼ Exp ( 2 ) ?
It turns out that the "Pocket Change" data set was taken from a random variable: X...
0.00 0.20 0.60 0.43 0.75 0.27 1.45 0.19 0.26 0.04 0.61 0.26 0.80 0.36 0.26 1.30 0.01 0.10 0.06 0.19 0.51 1.74 0.01 0.19 0.06 0.19 0.17 0.25 0.24 0.36 0.15 0.36 0.07 1.18 0.83 0.23 0.06 0.01 0.05 0.78 1.14 0.41 0.38 0.40 1.46 0.07 0.63 0.91 0.21 0.75 0.19 0.59 0.12 1.12 0.97 0.02 0.04 0.38 0.56 0.33 0.16 0.16 0.29 0.24 0.02 0.98 0.44 0.19 0.63 0.59 0.50 0.40 1.56 0.12 3.17 0.13 0.41 0.11 0.05 0.59...
Romeo and Juliet have a date at a given time, denote that random variable X and Y is the amount of time where Romeo and Juliet are late respectively. Assume X and Y are independent and exponentially distributed with different parameters λ and μ, respectively. Find the PDF of X – Y.
Consider an exponentially distributed random variable X with pdf f(x) = 2e−2x for x ≥ 0. Let Y = √X. a. Find the cdf for Y. b. Find the pdf for Y. c. Find E[Y]. If you want to skip a difficult integration by parts, make a substitution and look for a Gamma pdf. d. This Y is actually a commonly used continuous distribution. Can you name it and identify its parameters? e. Suppose that X is exponentially distributed with...
Recall from class that the standard normal random variable, Z, with mean of 0 and stan- dard deviation of 1, is the continuous random variable whose probability is determined by the distribution: a. Show that f(-2)-f(2) for all z. Thus, the PDF f(2) is symmetric about the y-axis. b. Use part a to show that the median of the standard normal random variable is also 0 c. Compute the mode of the standard normal random variable. Is is the same...
Problem 5 of 5Sum of random variables Let Mr(μ, σ2) denote the Gaussian (or normal) pdf with Inean ,, and variance σ2, namely, fx (x) = exp ( 2-2 . Let X and Y be two i.i.d. random variables distributed as Gaussian with mean 0 and variance 1. Show that Z-XY is again a Gaussian random variable but with mean 0 and variance 2. Show your full proof with integrals. 2. From above, can you derive what will be the...
A random sample of 30 was taken from the random variable X with pdf f(x)=1/2 on the interval [-1,1]. a) µ= b) σ^2 = b)Use the central limit theorem find p(0≤µ≤ 1/5 )approximately.
X is a random variable exponentially distributed with mean Y, where Y is uniformly distributed on the interval [0,2], Find P(X>2|Y>1) roblems: X is a random variable exponentially distributed with mean Y, where Y is uniformly distributed on the interval [0,2], Find P(X>2|Y>1) roblems:
Let X1, X2, ..., Xn be independent Exp(2) distributed random vari- ables, and set Y1 = X(1), and Yk = X(k) – X(k-1), 2<k<n. Find the joint pdf of Yı,Y2, ...,Yn. Hint: Note that (Y1,Y2, ...,Yn) = g(X(1), X(2), ..., X(n)), where g is invertible and differentiable. Use the change of variable formula to derive the joint pdf of Y1, Y2, ...,Yn.
Question 3 [17 marks] The random variable X is distributed exponentially with parameter A i.e. X~ Exp(A), so that its probability density function (pdf) of X is SO e /A fx(x) | 0, (2) (a) Let Y log(X. When A = 1, (i) Show that the pdf of Y is fr(y) = e (u+e-") (ii) Derive the moment generating function of Y, My(t), and give the values of t such that My(t) is well defined. (b) Suppose that Xi, i...
problem 3 and 4 please. 3. Find the moment generating function of the continuous random variable & such that i f(x) = { 2 sinx, Ox CT, no otherwise. 4. Let X and Y be independent random variables where X is exponentially distributed with parameter value and Y is uniformly distributed over the interval from 0 to 2. Find the PDF of X+Y.