# Questions tagged [density-function]

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

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### 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 ...
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### 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{...
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### 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.
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### 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 $a$ ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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
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### 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 $f_X$, $f_Y$ ...
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### 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 \...
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### 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 ...
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### 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,...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...