Questions tagged [density-function]

For questions on using, finding, or otherwise relating to probability density functions (PDFs)

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4
votes
2answers
67 views

Finding the probability density function of the $n$th largest random variable.

Let $X_1,...,X_{25}$ be independent Unif $[0,1]$ random variables. Let $Y$ be the $13$th largest of the $25$ random variables. Find the probability density function of $Y$. I already know the answer ...
0
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1answer
32 views

Why the median p.d.f. of the uniform distribution is not a p.d.f?

Let $X$ be uniformly distributed on interval $[\theta-2, \theta+2]$, $\theta\in\mathbb{R}$. Let the sample size of $3$, find the p.d.f. of median! I have tried as follows. The p.d.f. of $X$ is \begin{...
-3
votes
1answer
30 views

Simulate random vector [closed]

I would like to know how can I generate a random vector from this density function in Matlab $$f(x,y)=\frac{2}{(b-a)b}\mathbb{1}_{a<x<y<b}$$ Thank you very much.
0
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1answer
56 views

Finding the conditional distribution of Poisson random variables [duplicate]

Question Let $X$ and $Y$ be the number of accidents which will occur at each of two intersections over the next year. Suppose that $X$ and $Y$ are independent Poisson random variables , with means $a$ ...
0
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1answer
28 views

Probability density function problem on finding $\alpha,\beta >0$

Math isn't my strong suit and my professor just reads us the textbook for lectures so I apologize if this problem seems super simple. I'm just completely lost. Find $\alpha,\beta > 0$ such that $$f(...
3
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0answers
31 views

95 % two-sided confidence interval for Weibull distribution

I got this question: The Weibull distribution with $\alpha>0$ has density function: $$f_{\alpha}(x)=\alpha x^{\alpha-1}e^{-x^{\alpha}}$$ with respect to standard Lebesgue measure. X follow this ...
0
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1answer
37 views

What is the marginal pdf of V

Let X,Y be independent $Exp(\lambda)$ random variables. Let $U=X+Y$ and $V=X-Y$. Find the joint pdf of U,V then use it to find the marginal pdf for $V$. My attempt: Using $f_{U,V}(u,v)=f_{X,Y}(\gamma(...
2
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1answer
67 views

Find $\mathbb{P}(X<|Y|/2)$

The joint density of $X$ and $Y$ are given by $$f(x,y)=C(y-x)e^{-y},\quad-y<x<y,\quad 0<y<\infty$$ Find C Find the pdf for $X$. Find the pdf for $Y$. Find $\mathbb{P}(X<|Y|/2)$ My ...
-2
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1answer
17 views

density function of chi square random variables.

Supose that we have two random variables $X \sim \chi^2(n)$ and $Y \sim \chi^2(k)$. Let Z = X + Y. Without using Moment Generating Functions show, that $Z \sim \chi^2(n+k)$ Hint: Use two-dimensional ...
1
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2answers
26 views

PDF of the exponential of the sum of $N$ independent gaussian random variables

Is it true that the PDF of the sum of independent gaussian and identical random variables is gaussian with mean $N\cdot\mu$ and variance $N \cdot \sigma^2$? And if so how do we get the PDF of the ...
0
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1answer
34 views

Product of two densities

Let $X \sim \Gamma(\alpha, \lambda)$ and $Y \sim \Gamma(\beta, \lambda)$. I denoty by $f_X$ the density of X and by $f_Y$ the density of Y. Additionally, I assume that the density of (X, Y) is $f_X \...
1
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2answers
73 views

Most powerful test. Help using Neyman Pearson

$X_1,X_2,...$ independent continuous random variables with p.d.f $f(x) = \theta x^{\theta-1}$ if $0<x<1 , 0 $ otherwise for $\theta > 0 $ sample size = 1 Use Neyman-Pearson Lemma to drive MP ...
0
votes
1answer
31 views

Given density, different random variables, same distribution

Let $(X, Y)$ be a random variable in $\mathbb R^2$ with density $$f_{X, Y} (x, y) = \dfrac{\alpha (\alpha + 1)}{(1+x+y)^{2 + \alpha}} \mathbb 1_{[0, \infty)} (x) \mathbb 1_{[0, \infty)} (y)$$ for some ...
1
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3answers
36 views

Finding the probability density function of $X=2T_1+T_2$.

I am finding the probability density function $$ X=2T_1+T_2.$$ $T_1,T_2$ are independent with density function of $$f(t)=e^{-t}, t>0.$$ So what I did is find the $f(x)$ by using this integration: $$...
2
votes
1answer
48 views

Proving that a function is a probability density function

Let $(X_1,X_2,X_3)$ be a random vector with joint density $$\begin{cases} 8x_1x_2x_3, & x_1,x_2,x_3 \in (0,1) \\ 0, & x_1,x_2,x_3 \in (0,1)^c \end{cases}$$ Let $Y_1 =X_1$, $Y_2=X_1X_2$ and $...
0
votes
1answer
46 views

Most powerful test - hypothesis

sample of size n = 1 from p.d.f. $f(x| \theta) = 1 +\theta^2 (0.5-x)$ if $0<x<1, 0 $ o/w where $-1 ≤ \theta ≤ 1$ Derive the MP test for testing $$H_0 : \theta = 0$$ $$H_A : \theta = \theta_1$$ ...
1
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1answer
112 views

On the Wigner semicircle distribution

Question Let $(X, Y)$ be a jointly continuous pair of random variables such that the marginal density of $X$ is $$f_X(x) = \frac 1 {2\pi} \sqrt{4 - x^2}, \quad -2 < x < 2$$ (also known as the ...
0
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0answers
11 views

Find the maximum likelihood of k [duplicate]

A population has a density function given by f(x)=(k+1)x^k 0<x<1 0 otherwise For n observations,x1,x2,x3...xn made from this population, find the maximum likelihood of k. Here's my attempt: I ...
0
votes
1answer
20 views

Sum of a normally distributed rv and a Bernouilli distributed rv

I a trying to understand the following result from this lecture notes. Defining the value at risk as a function of the random variable $L$ and the parameter $\alpha \in (0,1)$: $$ \operatorname{VaR}_{\...
-1
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1answer
96 views

What is the probability density function of $Y=Z_1^2+…+Z_k^2$?

Let $Z\sim \mathcal{N}(0,1)$. What is the probability density function of $X=Z^2$? Let $Z_1,Z_2,...,Z_k\sim\mathcal{N}(0,1)$ be $k$ independent standard normal random variables. What is the ...
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0answers
142 views

Possible distribution of a specific random variable in joint probability density function [closed]

Let X1,...,Xn be i.i.d. continuous r.v. with a positive continuous joint probability density function f(x1,...,xn) a) Suppose that the distribution of X1,...,Xn is radially symmetric about the origin, ...
0
votes
1answer
63 views

Find integration limits of Probability distribution

I am trying to figure out limits of integration, to find the probability distribution of $P(X<Y-Z)$. Please verify and correct if I can write like below. $$P(X<Y-Z)=\int_{y=0}^{\infty}\int_{z=0}^...
1
vote
1answer
250 views

Finding all possible distributions of a continuous random variable

Question Suppose that a random variable $X$ has the property that, for any $a > 1$, the conditional distribution of $\frac 1 a X$ given that $X > a$ is the same as the distribution of $X$. ...
0
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0answers
35 views

Integration regarding joint probability density functions to find required expectations

Question Suppose that random variables $X$ and $Y$ have joint probability density function (PDF) $$f_{X, Y}(x, y) = e^{-x},\quad 0 < y < x.$$ Find $\mathbb{E}(Ye^{-X})$ and $\mathbb{E}(Ye^{-X}\ |...
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2answers
58 views

Integration regarding joint probability density functions to find required probabilities

Edit It seems I could not evaluate $\mathbb{P}(X + Y > 4)$ correctly as I had wrongly assumed that $4 - x$ is always positive, when in actual fact it could be negative, so there was another case to ...
1
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0answers
30 views

Probability density function of the mean of $n$ i.i.d random variables

Question: Assume that the $X_1,\dots,X_n$ are continuous i.i.d random variables. Calculate the probability density function of: $$Y_n:=\frac{1}{n}\sum_{i=1}^n X_i$$ My try: I thought maybe finding the ...
0
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0answers
8 views

Convex set and density estimation of Huber's contamination model

In the celebrated Huber's robust estimation paper, he considered the following model $x_i \sim (1-\epsilon) P_\theta + \epsilon G$ where $P_\theta$ is assume to be standard normal. Under this model, ...
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0answers
50 views

Finding pdf, E(Y) and sd for continuous random variable

Suppose the waiting time W for the outpatient clinic of a public hospital has a uniform distribution over the interval from 0 to 7 days. Researchers were interested in the the wait for the MRI was ...
0
votes
1answer
43 views

Symmetric PDF up to a proportionality

I have a random variable $X$ that has a pdf up to a proportionality defined by: $$f(x) \propto e^\frac{-(x-1)^2}{ 2}+e^\frac{-(x-4)^2}{2} \quad 0<x<5$$ Also is given that the $E(X)=2.5$.It seems ...
1
vote
1answer
43 views

Find pdf of $x+y$ uniform $(\theta, \theta +1)$?

How to find the pdf of $x+y$, where $x$, $y$ are iid and both are uniform $(\theta, \theta +1)$? I don't know how to divide intervals. I know how to solve if uniform $(0,1)$, since I can draw a square ...
0
votes
4answers
32 views

Determine the probability density function [closed]

Let X be a uniform random variable on the interval [1,3] and Y = X^2. How to determine the probability density function of Y
0
votes
1answer
16 views

Calculate density function $f_X$, $f_Y$

The random vector $(X,Y)$ has a density of $f_{X,Y} (x,y)=cxy I_D(x,y)$ where $D\subset \mathbb{R}^2$ the triangle with points $(0,0), (1,0), (1,1)$ and $c$ constant. Calculate $c$ and $f_X$, $f_Y$ ...
0
votes
1answer
27 views

Maximum likelihood estimation with uknown parameter [closed]

There is given an i.i.d with this realisation: 0.481, 1.612, 1.755, 1.077. The common density function of the above is: \begin{array}{ll} \frac{3x^2}{\vartheta^3} & \textrm{if } 0\leq x \leq \...
0
votes
0answers
9 views

Multivariate PDF Series Expansions

Suppose that I have a enough data that I can accurately estimate the higher order moments for a multivariate time series. (E.g. see Mardia's 1970 Measures of Multivariate Skewness and Kurtosis with ...
1
vote
0answers
17 views

Joint pdf of random variable and its function

Let be X an absolutely continuous random variable and Y=g(X) a second random variable (g is a Borel function $g:R\rightarrow R$). What’s the joint probability density function of the random vector (X,...
2
votes
1answer
39 views

How to calculate marginal density and identify which one they are?

Let $f_{UV}(u,v)=\frac{1}{2}\lambda_1\lambda_2 e^{-\frac{1}{2}\left((\lambda_1 +\lambda_2)u + (\lambda_2 - \lambda_1)v \right)}$ the joint density of U and V random variables. I have to decide if X ...
0
votes
1answer
22 views

Joint Probability Density Function from the graph [closed]

A random vector X=(X1,X2) is uniformly distributed in the grey region of the figure. Explain analytically the support and density function of X. How can I find the pdf by looking at this graph? Thank ...
0
votes
2answers
27 views

How do I bound the integrals for the probability density function?

probability density function is given by $f(x_1,x_2)=(-1.5)(1-x_2)$ if $0 \leq x_1 \leq x_2 \leq 2$. I need to find $P(x_1\leq 0.75, x_2 \geq 0.5)$. I'm not sure how to bound the double integral. Is ...
0
votes
1answer
53 views

Arrival Time for Poisson Process with density

Here is what I did to start this problem: Let A be the first attack. Let $X_1, X_2, ... $ be the i.i.d. arrival sequence of attacks. Then, $A = \sum_{i=1}^{k} X_i$ for each $k=1, ..., 5$. Let $Z_k$ be ...
0
votes
1answer
37 views

Density function of flip the coin twice

We have a rigged coin, the probability of obtaining heads is triple that of obtaining tails. Consider the variable $X$ defined as follows: We flip the coin twice in a row. If Heads are obtained on ...
0
votes
1answer
39 views

How to show that a function is a probability density function (regular normal distribution)?

I have that $f(z)=\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}z^{2}}$ which is stated to be the probability density function of a standard normal distribution on $\mathbb{R}$ and $f_{\alpha}(x|z)=\frac{1}{\...
0
votes
1answer
15 views

Joint Density of Bernoulli

Let $X_1$ and $X_2$ be iid $Bern(p)$. Let $f$ be the joint density of $(X_1,X_2)$, so $f(x_1,x_2)=P(X_1=x_1,X_2=x_2)$. What is $f(1,0)$? Since a bernoulli is $p$ at $1$ and $1-p$ at $0$, is $f(1,0) = ...
0
votes
1answer
47 views

Sum of two Gaussian PDFs

I want to find the sum of two univariate Gaussians with different variances. Considering that the sum of two independent normal variables $X_1$ and $X_2$ gives $X_1+X_2 ~ N(\mu_1+\mu_2, \sigma^2_1+\...
0
votes
1answer
43 views

Show property of exponential families

Show that if $X$ is a random variable with pdf in an exponential family, then $$\mathbb E\left[\sum_{j=1}^k\frac{\partial\phi_j(\theta)}{\partial\theta_m}t_j(X)\right]=-\frac{\partial \log c(\theta)}{\...
1
vote
1answer
33 views

Calculating the Expected Value of a Probability Density Function (steps)

I have this probability density function and I need to find its expected value: $$f(t)=be^{-bt}$$ Which was also given to be: $$E[X]=∫_{-∞}^∞tf(t)dt$$ $$E[X]=∫_{-∞}^∞tbe^{-bt}dt$$ I also know the ...
0
votes
1answer
22 views

Confused about marginal density

My book says the following If you have two continuous random variables $X$ and $Y$ in a joint pdf $f(x,y)$ then $f(y)$ = $\int_{-\infty}^\infty f(x,y)dx$ $f(x)$ = $\int_{-\infty}^\infty f(x,y)dy$ My ...
0
votes
1answer
51 views

Given $X\sim\text{Unif}(0,2\pi)$, find the probability density function of $Y = \sin(X)$

Given $X\sim\text{Unif}(0,2\pi)$, find the probability density function of the derived RV $Y = \sin(X)$. I attempted the equation as follows however I am confused how to get the range of the PDF as ...
2
votes
0answers
19 views

Estimate Expected Value with Density Bound

Summary: For a finite family of random variables $(X^{i})_{i\leq N}$, we have the density estimate. $$ p_t(y) \leq \int_0^\infty C\left(\frac{1}{\sqrt{t}}+e^{\displaystyle\epsilon y^2}\right)e^{\...
0
votes
1answer
45 views

Given $X \sim \exp(1)$ find the probability density function of $Y = g(X)$

Given $X \sim \text{exp}(1)$ find the probability density function of the derived RV $Y = g(X)$ where $Y$ is: $$ Y = \begin{cases} X & \text{when } X \leq 1 \\ \displaystyle \frac {1} {X} & \...
1
vote
0answers
29 views

Find the Probability density function of a mean random variable

lets say $Y_{1},\ldots,Y_{n}$ are simple random samples with the PDF: $f_{\theta}(y)=\theta y^{\theta - 1} \mathbb{I}(0 \le y \le 1) $ How can I find the PDF of $\bar{Y}$? is it even possible?

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