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Questions tagged [density-function]

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

A probability density function (PDF), or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value.

A random variable $X$ has density function $f_X$, where $f_X$ is a non-negative Lebesgue-integrable function, if:

$$Pr(a \le X \le b) = \int_a^b f_X(x) \, dx$$

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Joint PDF, given equation and points, find c. Integral setup

I hope everyone is having a great morning. My question here is about how the integrals were set up. Question: Find $c$ if $f_{X,Y}(x,y) = cxy$ for $X,Y$ defined over the triangle whose vertices are the points $(0,0),(0,1)$ and$…
Emad Zamout
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How can I compute the integral $e^{\left(\frac{x^2}{2}+xy-y^2\right)}$

Let (X, Y) be a pair of random variables with density $$f_{(X,Y)}(x,y):=\frac{1}{2\pi}e^{(\frac{x^2}{2}+xy-y^2)}, (x,y)\in\mathbb R.$$ Give the marginal laws of X and Y and identify it with usual laws. Are X and Y independent? I am stuck from the…
Gabriel Macotti
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Finding pdf of y when knowing pdf of x

I am trying to solve a problem which is quite easy but I just can't get the needed result. Task: $x$~${Exp}(1)$. Therefore, pdf: $f_x(x)=e^{-x}$ and $y=lnx$. I need to show that pdf of ${y=f_y(y)}=Exp(y-e^{y})$, y continues from minus infinity to…
lovetimberland
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conditional independence in an integral

Let $f(\cdot)$ denote a density. Measure theory aside, I know that $A$ and $B$ are conditionally independent given $C$ iff $f(a,b|c)=f(a|c)f(b|c).$ Also, $\int f(a,b|c)f(c) dc = f(a,b)$. But is it true that $\int f(a,b|c)f(c) dc = \int…
Jennifer
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Computing the value of constant $C$ and quartile Q2

I have a problem with computing the value of constant $C$ and quartile Q2. The task is that feature X has a density $f(x)$, $f(x) = -\infty < x < \infty$ , defined as: $$f(x) = \left\{ \begin{array}{ccc} x^3-x & \mbox{if} & x \in [-1,0] \\ C\cos(2x)…
Ganjira
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Sum of the reciprocals of the differences is $1$

We have distinct arithmetic progressions with common differences $d_1, d_2, \cdots,d_n$. If every natural number belongs to exactly one arithmetic progression, prove that $\frac{1}{d_1}+\frac{1}{d_2}+\cdots+\frac{1}{d_n}=1$. I saw two proofs of this…
green_blue
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density function of $Z=\sqrt{(X^2+Y^2)}$ when $f_{xy}(x,y)$ is given

A point is randomly chosen from a disk of radius R centred at the origin so that each point in the disk is equally likely than X and Y are jointly continuous with joint density given by $$f_{xy}(x,y)= \Biggr\{ \begin{array}{lr} …
Deepesh Meena
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