Linked Questions

2
votes
1answer
3k 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.
0
votes
1answer
556 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 ...
20
votes
3answers
7k 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\...
5
votes
2answers
523 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: $$f_X(x) =\begin{cases} a e^{-a x},& x\geq 0\\ 0,& \text{otherwise}.\end{cases}$$ ...
0
votes
2answers
6k 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+...
4
votes
1answer
733 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 ...
3
votes
1answer
387 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. ...
-1
votes
3answers
287 views

pdf for a difference of exponential random variables

Suppose $X$ is an exponential random variable with mean $2$, and $Y$ is an exponential random variable with mean $3$. I want to write the density function for $X-Y$. Now, I know (at least, I'm pretty ...
2
votes
1answer
229 views

Difference of Exponential Random Variables / Linear Transformations of RVs

Suppose X and Y are both distributed exponentially with parameter $\lambda$ and $\mu$ respectively. I am trying to find the distribution of X - Y via this method and it does not seem to be working, ...
1
vote
1answer
151 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 ...
1
vote
0answers
206 views

Expected value (waiting time) of two independent exponentially distributed arrival times occuring in a given distance

Suppose we have two independent exponentially distributed arrival times X1, X2 having rates λ and μ. This means their corresponding expected waiting times are 1/λ and 1/μ accordingly. Now I'm ...
0
votes
1answer
105 views

Is the distribution of one exponential will be smaller than a second one Uniform?

I came by an expression which I am not sure I understand. If: $X_1 \sim exp(\lambda)$ $X_2 \sim exp(\lambda)$ Then: $P(X_1<X_2|X_2) \sim Uniform(0,1)$ Where it is not clear to me what "$|X_2$...
0
votes
2answers
49 views

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)} &...
-1
votes
1answer
58 views

Question about exp. distribution [closed]

We know that $X\sim \exp(1),Y\sim \exp(2)$ and they are independent. What is $P(Y>X)$? exp=Exponential... Thank you!
1
vote
1answer
69 views

Write a conditional probability expression with CDF/PDF

When I read maths chapter, I found following conditional probability expression in a part of intermediate step and then directly gave the answer \begin{equation} I=\Pr\{x\leq y|x=t\} \end{equation} ...

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