# 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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### 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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1 vote
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• 1,096
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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 ...
• 2,382
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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 ...
33 views

### 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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### 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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### 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 ...
52 views

### 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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