Homework Statement
Consider the exponential probability density function with location parameter ##\theta## :
$$f(x|\theta) = e^{-(x-\theta)}I_{(\theta,\infty)}(x)$$
Let ##X_{(1)}, X_{(2)},...,X_{(n)}## denote the order statistics.
Let ##Y_i = X_{(n)} - X_{(i)}##.
Show that ##X_{(1)}## is...
Hello good people of PF, I came across this problem today.
Problem Statement
Given bivariate normal distribution ##X,Y \sim N(\mu_x=\mu_y=0, \sigma_x=\sigma_y=1, \rho=0.5)##,
determine ##P(0 < X+Y < 6)##.
My Approach
I reason that
$$ P(0 < X+Y < 6) = P(-X < Y < 6-X)$$
$$ =...
Hi everyone, I am currently working through the textbook Statistical Inference by Casella and Berger. My question has to do with transformations.
Let ##X## be a random variable with cdf ##F_X(x)##. We want to find the cdf of ##Y=g(X)##. So we define the inverse mapping, ##g^{-1}(\{y\})=\{x\in...
Thanks everyone for your help. I see now that the answer is a matrix with elements that are the covariance between ##AX^{(1)}## and the elements of ##BX^{(2)}##.
Homework Statement
Given random vector ##X'=[X_1,X_2,X_3,X_4]## with mean vector ##\mu '_X=[4,3,2,1]## and covariance matrix
$$\Sigma_X=\begin{bmatrix}
3&0&2&2\\
0&1&1&0\\
2&1&9&-2\\
2&0&-2&4
\end{bmatrix}.$$
Partition ##X## as
$$X=\begin{bmatrix}
X_1\\X_2\\\hline X_3\\X_4\end{bmatrix}...
##f_X(x) = 2\int_x^\infty e^{-x}e^{-y}dy = 2e^{-2x}##
##f_Y(y) = 2\int_0^y e^{-x}e^{-y}dx = 2e^{-y}(1-e^{-y})##
##f_X(x)f_Y(y) = 4e^{-2x}e^{-y}(1-e^{-y}) \neq f_{X,Y}(x,y)##
So we can show explicitly that they are not independent...
But we can separate ##f_{X,Y}(x,y)## into a product of...
Homework Statement
Given ##f_{X,Y}(x,y)=2e^{-x}e^{-y}\ ;\ 0<x<y\ ;\ y>0##,
The following theorem given in my book (Larsen and Marx) doesn't appear to hold.
Homework Equations
Definition
##X## and ##Y## are independent if for every interval ##A## and ##B##, ##P(X\in A \land Y\in B) = P(X\in...
This problem had 4 steps with equal probability. So in general with ##n## steps and ##p## probability, we have ## P(ending \ at \ w) = {{n}\choose{k}} p^k (1-p)^{n-k} \ where \ k = \frac {(n+w)} {2} ##?
Outcomes are [-4,4].
The variable k is the number of successes. Let's define success as +1 on the x axis.
The relation between k and the outcomes...
k=0 gives the outcome -4,
k=1 gives the outcome -2,
k=2 gives the outcome 0,
k=3 gives the outcome 2,
k=4 gives the outcome 4.
So ##P(w) =...
Homework Statement
Suppose a particle moves along the x-axis beginning at 0. It moves one integer step to the left or right with equal probability. What is the pdf of its position after four steps?
2. Homework Equations
Binomial distribution
##P(k) = {{n}\choose{k}} p^k (1-p)^{n-k}##
The...