Questions tagged [density-function]

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

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Variance for random variable with known density

let $t,h,g\in \mathbb{R}$ and $$h(z|x)=\frac{1}{\sqrt{2\pi}}e^{-1/2(z-t-hx-gx^2)^2}$$ $$f(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$$ and $g(x,z)=f(x)h(z|x)$. Find the variance of $Z$ I have found the ...
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Identifying the given PDF?

I came across the following expression of PDF of $n$ exponential random variables. $f_X(x) = \sum_{n=1}^N$ $\binom{N}{n}$ $\frac{n(-1)^{n-1}}{\Omega}\exp(\frac{-nx}{\Omega})$ My query is how this PDF ...
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Conditional density: exact definition

I have trouble understanding the connection of conditional densities for two (continuous) random variables and conditional distributions for measurable sets. Let $\nu$ be a probability measure on $\...
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A circle and the probability as a function of time

I am trying to solve the following problem.Supposing that we have a circle of radius R which at t=0 does not contain any sphere as we had set a border so that no sphere was allowed to enter in it.At t=...
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How to find PDF of a random variable when it is squared and multiplied by constant?

I am getting confused in the following situation. Say we have a random variable $X$ whose PDF is known to us. Now how to find the PDF of another random variable say $Y$, which is related to $X$ as $Y =...
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Clarification about inequality in summation

In my work I am facing the following situation, wherein I am trying to compute CDF of random variable $Y$ such that $F_Y(y) = \text{Pr}(\sum_{m = 1}^M |Z_m|^2\leq \frac{y}{A})$ -----(1) where $Z$ is a ...
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Understanding absolute value of random variables

In my work I am facing the following situation: $y = |a+bc|^2$ ----(1) where $a,b,c$ are zero mean circularly symmetric complex Gaussian (ZMCSCG) random variables with variance $\sigma^2_a, \sigma^2_b,...
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PDF of following random variable.

I am trying to find the PDF of random variable $X$ but not getting it correctly. $X = \sum_{m=1}^{N}|a_m+\eta b_m c_m|^2$ ----(1) where $a, b, c$ are Zero mean circularly symmetric complex Gaussian (...
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Calculating the transformation of 1/X where the pdf of X is 1/x

I feel very silly asking this, since it seems so simple but I'm almost certain I have done something wrong. I have a random variable $A$ with the following pdf: \begin{equation} f_{A}(x_{A}) = \begin{...
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Determining the pdf of the product of two independent random variables

I have two iid random variables $A_{1}$ and $A_{2}$ with a probability density function as such: \begin{equation} f(x) = \frac{1}{12x} \end{equation} In the domain $x \in [0.5, 0.6]$, and $0$ ...
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Data required for Dual Dirac PDF

I have a simple question regarding the dual Dirac PDF. If I have a set of deterministic data, e.g., d = [-2ps, 2ps, -2ps, 2ps, -2ps, 2ps] Would the resulting PDF look like a dual Dirac PDF? Where ...
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Autocorrelation of a random process

Let X be a random process. X(t) = A*Cos(wt+θ) ; where A and w are constants. The only random thing is θ. Lets say θ has a probability density function, f(θ)= 1/2pi for 0<θ<2pi and zero elsewhere....
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Why the function f(x) is not a probability density function(PDF)

I saw this example but I can't understand why its's not PDF. Exmaple: Suppose $ f(x)=\left\{\begin{array}{ll} c\left(1-x^{2}\right) & \text { if }-2 \leq x \leq 2 \\ 0 & \text { otherwise. } \...
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Monte Carlo estimation of this probability

Let $p=P(X+Y\geq t)$ and $t\in \Bbb R$. Question: Using the classic Monte Carlo method, find an estimator $p_n$ of $p$ using $F^{-1}_X$ and $F^{-1}_Y$ Attempt: I defined $$Z=X+Y$$ then I expressed $$p=...
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Natural Density For Subsets of $\mathbb{N}^2$?

Suppose that we have an set $S\subset\mathbb{N}^2$. Taking the natural order on $\mathbb{N}^2$ induced by $f((n,m))=$ max$(\{n,m\})$, what can we say about the "natural density" of S in $\...
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If the law of a random variable $X$ is zero at a point, is then the density function of $X$ also zero at that point?

Let $(\Omega, F, \mathbb{P})$ be a probability space, $X$ is a random variable s.t. $X[\Omega] = [-a, a), a > 0$ and that $X$ has a density function $f_X$. If we know that $\mathbb{P}_X(c) = 0$ for ...
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Second order Taylor expansion of $\int_0^\mu\frac{x}{2\pi} - F(x)dx$ when $F(x)$ is a distribution function

I'm currently reading an article in which the authors perform a second order Taylor series approximation $\int_0^\mu\frac{x}{2\pi} - F(x)dx = \frac{\mu^2}{4\pi} - \frac{\mu^2}{2}F'(0) + o(\mu^2)$ when ...
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$f_U(t)=f_{X,Z}(t,1)+f_{Y,Z}(t,0)$ if $U=X $ if $Z=1$ and $U=Y$ if $Z=0$

Suppose $Z$ is a bernouli variable with parameter $p$. $Z,X,Y$ are independent random variables. Now $U$ be a random variable such that $$U(\omega)=\begin{cases}X(\omega) & \text{when }Z(\omega)=1\...
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Distribution of $f(X)X$ when $X$ is Uniform?

Suppose that $f$ is the probability density function of a probability measure $P$ which is absolutely continuous with respect to Lebesgue measure. Suppose X is a uniform random variable on the ...
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How to calculate the mass of a 3-D sphere collapsed into 2-D plane?

My sphere has density, $p(R)=(R+10)^-2$. If I collapse this sphere into a $2D$ plane, let's say it forms a $2D$ ellipse as a result, how will I calculate the mass of this $2D$ ellipse? in what ...
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Find density functions of $Z=Y-X$ when the joint density function is known.

Find the density function of $Z=Y-X$ where the joint density function of X and Y is given by $f(x,y)=1/2,x>0,y>0,x+y<2$ and $0$ otherwise. I know how to do it by finding the CDF first with ...
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Simplifying integration over $\min\{\cdot\}$

One RV, one individual constraint Let $X$ denote a random variable with PDF $f : [0,\infty) \to \mathbb R_+$ and constraint $x \leq c$. Consider the following integral \begin{align} g(c) = \int\...
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Evaluating the polar coordinate integral $\int e^{k\mu^Tx}dx$ when $c(k)\int e^{k\mu^Tx}dx = 1$ and $x \in \mathbb{S}^{p-1}$

Let $x \in \mathbb{S}^{n}$ be a point on the unit $n$-sphere with coordinates $\begin{cases} x_1 &= \cos(\theta_1).\\ x_2 &= \sin(\theta_1)\cos(\theta_2).\\ x_3 &= \sin(\...
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Find the value of a for valid PDF

I'm confused as how "a" has to do with anything with the PDF as it's not on the RHS?
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Finding the CDF of a Discrete RV P[X=a/n]

I am trying to find a CDF for this PMF: \begin{equation} \label{eq1:givenpdf} \mathbb{P}\left[X=\frac{a}{n}\right] = \frac{36}{5}\frac{n^2}{\left(n+1\right)\left(n+2\right)\left(n+3\right)\left(n+...
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Why can we use this density for random variables with normal distribution?

We have given $N\sim \mathfrak{N}(0;1)$, $x\in \Bbb{R}$ and $\epsilon>0$. Furthermore $f$ is continuous and bounded. Then we want to compute $E(f(x+\epsilon N))$ We have just proven that for a ...
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Is there a random variable which has a density but the distribution function is not differentiable?

I want to check or find a counterexample for the following statement The distribution $P_X$ of a random variable $X$ has a density iff it's distribution function $F_X$ is continuously differentiable. ...
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Conditional PDF of X given that Y>0

I have this joint distribution $f_{X,Y}(x,y) = \frac{1}{2}I_A(x,y)$ $A=\{(x,y) \in R^2; -1 < y < 1; 0 < x < 1\} $ What I need is to find the conditional distribution of X for when Y>0 $...
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Given independent $Z_i \sim N(\mu_i, \sigma_i ), i=1,2$ derive the density of $(Z_1, Z_1 + Z_2)$.

Given independent $Z_i \sim N(\mu_i, \sigma_i ), i=1,2$ I would like to derive the distribution of $(Z_1, Z_1 + Z_2)$ and doing so through deriving a density (but not using Characteristic functions). ...
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Proving the set where probability density function becomes infinite is bounded

For a continuous random variable $X$, with probability density function $p_X(x)$, it is known that there exists a $p_{min} > 0$ such that $p_X(x) \geq p_{min} \forall x \in X$. Also, I know that $X$...
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Joint distribution of the Sum of gaussian random variables

Suppose $X_1,X_2,X_3$ are iid with distribution $\mathbb{N(\theta, \sigma^2)}$ and $Y_1 = X_1 + X_2$ and $Y_2 = X_2 + X_3$. I need to find the joint distribution of $Y_1, Y_2$. Here is my attempt: ...
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Compute the density function of a pushforward measure

Problem Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ with $f(x,y) = (e^{2x+y}, e^{x+y})$. Compute the density function $\frac{df[\lambda_2]}{d\lambda_2}$ of the pushforward measure $f[\lambda_2]$, where $\...
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How to prove that probability density function of the sum of 2 independant variable is equal to their convolution?

The probability density function of the sum of two independent random variables is the convolution of their individual probability density functions. What is the simplest demonstration that proves ...
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Derive density of $Z=XY$, $X\sim U(0,1)$ and $Y\sim\mathcal N(0,1)$.

I have been stumped for a few days on this. I have two random variables $X\sim U(0,1)$ and $Y\sim\mathcal N(0,1)$, which are independent. How can I get the density of $Z = XY$? I put $Z = XY, W = Y$ i....
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CDF and cumsum of a prbabolity density function

This is a really really stupid question, but why when I plot a CDF and cumsum of a PDF (e.g. exponential): $$ f(t) ~ = \lambda e^{-\lambda t}, ~~~ t \ge 0 $$ I get ...
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$X,Y$ independent r.v. uniformly distributed on $[0,1]$. Find $\mathbb{P}(Y\le \frac X2)$

I'm new to these concepts, I have two approaches wich gives the same result, but I don't know if they either are both right or not. First: We must have $\mathbb{P}(Y=\frac X2)+\mathbb{P}(Y<\frac X2)...
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Finding constant in probability density function

I'm new to this course, I googled similar problems and watched several videos on how to solve similar problems but I'm still not sure how to solve this problem. As far as I understand the total area ...
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integral of $xf(x)$ for normal distribution

from Gregory book i read that $\int_{-\mu/\sigma}^{\infty}(\mu + \sigma x)f(x)dx = \mu F(\mu/\sigma) + \sigma f(\mu/\sigma)$ where $f(x)$ is the density of a normal distribution and $F(x)$ the ...
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Is every radon-nikodym derivative a random variable?

Let $(\mathsf{X}, \mathcal{X}, \mu)$ be a measure space with $\mu$ begin $\sigma$-finite. Definition of Random Variable: Let $(\mathsf{Y}, \mathcal{Y})$ be a measurable space. A function $\xi:\mathsf{...
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Showing a kernel density estimate with Gaussian Kernels is a probability distribution

Using a definition similar to the wikipedia definition here: Suppose that $(x_1, \ldots, x_n)$ are i.i.d samples from some univariate distribution with an unknown density $f$ at any given point. Then ...
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1 answer
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Distribution of Y=|X| [closed]

X ~ f(x) = K|x|, -1<x<2. Derive the distribution of Y=|X| and find the value of K. I have proceed the problem like this, but I don't know how to solve this problem. Please help me with this. Y=|...
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1 vote
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Density of a random variable under non-bijective map

Let $X$ be a continuous random variable with continuous density $f_X$ and $\phi$ a bijection that behaves well (i.e. with compatible assumptions). The density of $f_{\phi(X)}$ is given by : $$f_{\phi(...
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Calculating the joint cumulative distribution function from a junction tree

Assume I have the following Junction Tree between random variables $X_1,\dots,X_7$ that exactly describes the sets variables with non-zero Mutual Information (Alternatively it's the last tree in a ...
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Density of Final zero sequences

Ive got this situation We have the final zero sequences C_00 and we have to prove that is dense in C_0. The idea is proving that the clausure of C_00 is C_0. And the C_00 is defined by the sequences ...
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Represent $(Z_1,Z_2) \sim BVN(0,0;1,1;\rho)$ as a transformation of $W,W_1,W_2 \stackrel{i.i.d.}{=} N(0,1)$

Using the "transformation method", I am to show that with $W,W_1,W_2 \stackrel{i.i.d}{=} N(0,1)$, a random variable $Z \sim BVN(0,0;1,1;\rho)$ is identically distributed to the random ...
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Integration of $n$ isosceles triangles

Think of $n$ isosceles triangles with area $1$. All triangles look like the same. Each of them can be considered as a $f_{i}$ density function located somewhere on the real line, where the base of the ...
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Joint Density Function of two uniform distributions with two discrete cases

Let there be two random variables $X_1$ and $X_2$. With probability $\gamma$, they are perfectly correlated and each distributed uniformly on the interval $[0,1]$. With probability $1-\gamma$, they ...
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Proving Independence with 3 different variables, but only 2 different equations

If there are 3 i.i.d variables (X1, X2, and X3) with the same marginal probability density functions, but only 2 equations (Y1 and Y2) that contain some combination of those 3 variables, would it be ...
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1 answer
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distance between histogram and density in R [closed]

I have some financial data, here I have the histogram and density with estimated parameters regarding normal distribution. What I want now is to determine the "distance" between the data of ...
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3 votes
1 answer
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Integrating density over measurable sets yields probability?

I have a bit of trouble proving a statement expressing the expectation of a function of random variables as the conditional expectation, namely: $$\mathbb{E}g(X,Y) = \int_{\mathbb{R}^n} \int_{\mathbb{...
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