Linked Questions
35 questions linked to/from Pdf of the difference of two exponentially distributed random variables
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Difference of two exponential RVs [duplicate]
Let $X,Y$ be independent exponential RV's with respective pdf's $f(x) = \lambda e^{-\lambda x}$ and $f(y) = \mu e^{-\mu y}$. We want to find the pdf of $Z=X-Y$.
I originally tried the convolution ...
- 1,457
2
votes
1
answer
5k
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Linear Combination of Exponential Random Variables [duplicate]
Let $Y \sim \exp(\delta)$ and $T \sim \exp(\lambda)$, and $Y$ and $T$ are independent. How do I get the density $f(x)$ where $X=Y-cT$, $c>0$? Thanks.
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1
answer
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Finding the probability density function for IID rv [duplicate]
The question is as follows:
Suppose that X1 and X2 are independent, identically distributed exponential random variables. Determine the PDF for for X1 - X2.
I understand that because X1 and X2 are ...
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0
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2
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How to Find CDF of $T_1-T_2$? [duplicate]
Given that $T_1, T_2$ are iid $\text{exp}(\lambda)$ variates.
I want to find the cdf $F_T(t)$ where $T=T_1-T_2$
My Attempt
$F_T(t) =_1 P(T<t) = P(T_1-T_2<t) = P(T_1<T_2 + t)$
Where $=_1$ is ...
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0
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Find PDF of $Y = X_1-X_2$ [duplicate]
$X_1$ and $X_2$ are i.i.d random variables and the pdf of each of them is $e^{-x}$ for $x>0$ and $0$ otherwise. $Y = X_1-X_2$ and the question asks to find the pdf for $Y$? I took the approach of ...
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How to deduce the CDF of $W=I^2R$ from the PDFs of $I$ and $R$ independent
Given pdf of $I$ and $R$ (both $I$ and $R$ are independent RV's), how to find cdf of $W =I^2R$?
Where,
$$
\begin{align}
f_I(i)&=6i(1-i), &0 \leq i \leq 1 \\
f_R(r)&=2r, &0 \leq r\...
user754
5
votes
5
answers
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Density of $X-Y$ where $X,Y$ are independent random variables with common PDF $f(x) = e^{-x}$?
$X,Y$ are independent random variables with common PDF $f(x) = e^{-x}$ then density of $X-Y = \text{?}$
I thought of this let $ Y_1 = X + Y$, $Y_2 = \frac{X-Y}{X+Y}$, solving which gives me $X = \...
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2
answers
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Prove that $\mathbb P(X>Y) =\frac{b}{a + b}$ if $X, Y$ are exponentially distributed with parameters $a$ and $b$.
Let $X, Y$ be an exponentially distributed random variables with parameters $a, b$. Then $X$ has pdf:
$$f_X(x) =\begin{cases} a e^{-a x},& x\geq 0\\
0,& \text{otherwise}.\end{cases}$$ ...
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2
answers
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Exponential Random Variables
QUESTION:
Let $X$ and $Y$ be exponentially distributed random variables with parameters $a$ and $b$ respectively. Calculate the following probabilities:
(a) $P(X>a)$
(b) $P(X>Y)$
(c) $P(X>Y+...
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1
vote
1
answer
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Probability with exponential random variable
Machine $1$ is currently working. Machine $2$ will be put in use at
time $t$ from now. If the lifetime of machine $i$ is exponential with
rate $\lambda_i=1,2$, what is the probability that machine ...
- 3,125
3
votes
1
answer
993
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Probability of a sequence of events in a Poisson process.
I am starting to study Poisson processes and I came up with this question:
Let there be two Poisson processes with rates $\lambda$ and $\mu$ respectively, monitoring the occurrence of events (e.g. ...
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1
vote
1
answer
2k
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Difference of two exponential distribution
Let $X\sim Exp(\lambda_1)$ and $Y\sim Exp(\lambda_2)$, with $Z = X - Y$.
I am trying to find the pdf of Z, i.e. $f_Z(z)$.
Here is what I have got:
\begin{align*}
f_Z(z) &= \int_0^{z}...
- 95
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votes
2
answers
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PDF of |X-Y| where X,Y ~ exp($\lambda$)
I need to find the PDF of $Z=|Y-X|$ given that $X,Y\sim exp(\lambda)$ and both independent.
What I did (I want to use the CDF and not convolutions):
$f_{X,Y}(x,y)=\cases{\lambda^2 e^{-\lambda(x+y)} &...
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3
votes
2
answers
415
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$X \sim \exp(1)$, $Y \sim \exp(1)$ Independent what is the CDF of $Z=X-Y$?
Let $X \sim \exp(1)$, $Y \sim \exp(1)$ $Z=X-Y$.
$X,Y$ are independent.
What is the distribution of Z?
For $t\geq0$, I simply calculated that the old fashioned way.
$F_Y(t)=P(Z\leq t)=P(X-Y\leq t)=\...
- 929
2
votes
2
answers
642
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Probability machine 1 is the first to fail
We are given two machines call them $M1$ and $M2$. $M2$ will be put in
use at a time $t$ from now. The lifetime of machine $i$ is exponential
with rate $\alpha_i$ $i=1,2$. What is the probability that ...
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