Linked Questions

2 votes
3 answers
4k views

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 ...
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  • 1,419
2 votes
1 answer
4k views

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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  • 21
0 votes
1 answer
809 views

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 votes
2 answers
82 views

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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  • 61
0 votes
0 answers
127 views

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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23 votes
3 answers
10k views

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\...
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5 votes
5 answers
3k views

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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  • 5,484
6 votes
2 answers
851 views

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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  • 721
0 votes
2 answers
7k views

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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  • 299
1 vote
1 answer
2k views

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 ...
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  • 2,975
3 votes
1 answer
840 views

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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  • 273
1 vote
1 answer
1k views

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}...
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3 votes
2 answers
266 views

$X \sim \exp(1)$, $Y \sim \exp(1)$ 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)=\...
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2 votes
1 answer
999 views

Waiting time: exponential distribution

Smith is waiting for his two friends Lee and Yang to visit his house. The time until Lee arrives is Exp($\lambda_1$) and the time until Yang arrives is Exp($\lambda_2$). After arrival, Lee stays an ...
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  • 513
2 votes
2 answers
451 views

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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  • 3,819

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