Linked Questions
17 questions linked to/from X,Y are independent exponentially distributed then what is the distribution of X/(X+Y)
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Finding density of $U = \frac{X}{X + Y}$ for $X, \ Y $ ~ $\text{Exp}(\lambda)$ i.i.d [duplicate]
Problem:
Given $X, Y$ ~ $\text{Exp}(\lambda)$ i.i.d, find $f_U, \ F_U$ for $U := \frac{X}{X + Y}$.
My approach:
For a fixed $u > 0$, parameterize $\{ (x,y) | \frac{x}{x + y} = u \}$ = $\{ (x,y) | y ...
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0
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2
answers
4k
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If $(X,Y)$ are i.i.d. exponential then $X/(X+Y)$ follows a Beta distribution [duplicate]
I'm asked to show that if $X$ and $Y$ are independent exponential random variables with parameter , then has a Beta distribution.
Up to know I had to find the new pdf or df when $Y$ was of the kind $...
- 103
1
vote
1
answer
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$X,Y$ are independent exponentially distributed: What is the distribution of $X/(X+Y)$ [duplicate]
I'm a business major, so really distribution theory is not my strong suit. I know how to get the distribution of a ratio of exponential variables and of the sum of them, but i can't piece everything ...
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0
votes
1
answer
808
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Show that random variable $U=\frac{X}{X+Y}$ has uniform distribution on [0,1] when X & Y are independent random variables with same exp. distribution. [duplicate]
Show that the random variable $U=\frac{X}{X+Y}$ has a uniform distribution on the range [0,1] when X and Y are independent random variables with the same exponential distribution.
I'm stuck at
$F_U\...
- 317
0
votes
1
answer
328
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Transformation and marginalization of a joint distribution function [duplicate]
Suppose that the joint probability density function of X1 and X2 is given by:
$$f(x_1; x_2) = exp(−x_1 − x_2)$$ $x_1 > 0$; $x_2 > 0$ and $0$ $elsewhere$. We define Y1 and Y2 as follows: $$Y_1 = ...
- 481
18
votes
3
answers
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How exactly are the beta and gamma distributions related?
According to Wikipedia, the Beta distribution is related to the gamma distribution by the following relation:
$$\lim_{n\to\infty}n B(k, n) = \Gamma(k, 1)$$
Can you point me to a derivation of this ...
- 183
3
votes
2
answers
2k
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Show that the random variables $Y_1$ and $Y_2$ are independent
Let $X_1,X_2$ be i.i.d with pdf
$$f_X(x)=\begin{cases}
e^{-x} & \text{for } 0< x<\infty{}
\\0 & \text{elsewhere }
\end{cases}$$
Show that the random variables $Y_1$ and $Y_2$ with $...
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5
votes
1
answer
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Show that two random variables are independent
I'm learning probability and need help with the following problem :
Let $X_1, X_2$ be independent and identically distributed random variables with probability density function
$$f(x_i) = \...
user347616
4
votes
1
answer
1k
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How to prove these two random variables are independent?
If $X$ and $Y$ are independent Gamma random variables with parameters $(\alpha,\lambda)$ and $(\beta,\lambda)$ respectively, how to show that $U=X+Y$ and $V=X/(X+Y)$ are independent?
- 5,287
0
votes
2
answers
2k
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What's the probability of truck A arrives before truck B [closed]
Truck A arrives at a random time between 9am and 11am, and truck B arrives at a random time between 10am and 12pm (noon). what are the odds that truck A arrives before truck B??
I searched this ...
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6
votes
1
answer
383
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Sufficient conditions for $X/(X+Y)$ to have a uniform distribution
Suppose that $X$ and $Y$ are i.i.d. r.v.'s with an exponential distribution of parameter $1$.Then it is known that the ratio
$$Z = \frac{X}{X+Y}$$
has a uniform distribution on $(0,1)$. See for ...
- 5,508
1
vote
1
answer
345
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Is it possible to use MGFs for find the distribution of $X/(X+Y)$ when $X$ and $Y$ are independent and gamma distributed?
Suppose that
$$Z=\dfrac{X}{X+Y}$$
$$X \sim Gamma(a,\lambda)$$
$$ Y \sim Gamma(b,\lambda)$$
with $X$ and $Y$ independent. I would like to see if it might be possible to determine the distribution of $...
- 2,624
0
votes
1
answer
179
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$X_1,X_2$ independent gamma-distributed random variables. Density of $Y:=\frac{X_1}{X_1+X_2}$
Let $X_1,X_2$ be two independent gamma-distributed random variables: $X_1 \sim \Gamma(a_1,b), X_2 \sim \Gamma(a_2,b)$.
How can I determine the density of $$Y:=\frac{X_1}{X_1+X_2}$$
I don't really ...
- 215
0
votes
0
answers
179
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How do you generate 10 values from a beta distribution?
I want to generate 10 values from a beta distribution on the interval [0,1] using these parameters β_1=1⋅47 and β_2=2⋅16.Next transform them to be on the interval [-10,20].
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2
votes
1
answer
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What is the distribution of $PX/(PX+(1-P)Y)$ if $P$ is uniform and $X,Y$ are exponential?
Suppose $X,Y$ are exponentially distributed with $\lambda =1$ and $P$ is uniformly distributed on $(0,1)$. All random variables are independent. What is the distribution of
$$Z=\frac{PX}{PX+(1-P)Y}?$...
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