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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:

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

2263 questions
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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 define $g(x) = \frac{1}{n} \sum_{i=1}^{n}…
IntegrateThis
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Conditional Density

Question: X follows uniform [0, 1] and Y|X follows uniform[0,X]. What is the distribution of X|Y? My Try: $$f_{Y|X}(y|x) = \frac{f_{X,Y}(x,y)}{f_X(x)} = \frac{1}{x}, \, with\, f_X(x)=1$$ $$f_{X|Y}(x|y) = \frac{f_{X,Y}(x,y)}{f_Y(y)}$$ I am not sure…
quant_to_be
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Probability density function maximum area under the curve

For a probability density function with min and max limits on $x$, Is it correct to say that the area under the curve between those limits must equal $1$? E.g. $f(x) = kx^2 \quad 0 \le x \le 1$ So $k$ must be set to a value to ensure the area is…
Bryon
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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 question is, is this by definition or is there a…
user911315
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How to find the marginal density $f(y)$ for the following figure?

The figure is in the above image. I am told that the joint density $f(x,y)$ is constant in the region shown in the figure. So $f(x,y) = 1/12$ for the region of the figure. I am asked to find $f(y)$ and $f(x|y)$ I need help finding $f(y)$ in order…
user908519
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Determining the constant $k$ of a density function

The diameter of a grain of sand in a sand roof, measured in mm, can be considered as a random variable X with probability density $f(x) = k(x-x^2)$ for $0\leq x \leq 1$ $f(x) = 0$ otherwise Decide the constant $k$ such that this becomes a legitimate…
Mampenda
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How to find density of one random variable if joint pdf contains $3$ random variables?

My book asks me the following question: Let $X, Y, Z$ have joint density $f(x, y, z) = 6$, for $0 < x < y < z < 1$ and $f(x, y, z) = 0$ otherwise. a. Are $X$ and $Y$ and $Z$ independent? I wrote no because it doesn't seem like the region of $0 < x <…
user865043
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How to calculate density functions from joint pdf with the following constraints?

Let $X,Y$ have joint density $f(x, y) = 3xy/1250$ for $0 ≤ x, 0 ≤ y, x + y ≤ 10$ and $f(x,y) = 0 $ otherwise Find the density $f(x)$ of $X$. My attempt: $f(x) = \int_0^{10-x} \cfrac{3xy}{1250}dy = \cfrac{30}{1250}x^2-\cfrac{3}{1250}x^3$ for $0\leq x…
user865043
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Find the probability density function for 2 random variables

I have a question and I will be happy to get some help. $X$ and $Y$ are random variables that are distributed uniformly in $(0,1)$ (Continuous Uniform distribution). EDIT: X and Y are independent $W$ is defined as $X^2 - Y$. What is the probability…
Lidor Hacham
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Is $f_{XY}(x,y)$ the same as $f(x,y)$?

I have a slight confusion with the notation, sometimes I find the joint density function like this: $$f_{XY}(x,y)$$ and on other occasions like this: $$f(x,y)$$ Is the same or what is the difference? Thanks!
JanetN
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Do densities lie in $L^p$?

Let $f$ be a probability density. We know $f \in L^1$ since by definition, it integrates to 1. But is $f \in L^p$ for any $p > 1$? It seems logical to assume so? We know that densities approach zero for $|x| >> 0$. So if you square it, or take it to…
saei
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density function and measure null

If $P = fd\mu $, $Q = gd\mu$ and $P = h dQ$ do we have $$ \mu(g=0)=0 $$ My goal is to show that $h = \frac{f}{g} $ almost $\mu$ everywhere. thanks and regards.
user347910
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Casorati-Weierstraß theorem - dense

I try to understand the Casorati-Weierstraß theorem. But I don't understand when a picture is dense in C. $e^{1/z}$ is dense, $1/z$ isn't. But why? Thanks.
malilini
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probability from joint density function

Could anyone help with this problem? Thanks A joint density function is given as follows: $$f(x,y) =\begin{cases}{} 0.125\cdot (x+y+1) \ \ \text{for} -1Y)$
Denson
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The joint density of random variables $X$ and $Y$ is given by $P(X=n, Y=m) =$...

The joint probability of the random variables $X$ and $Y$ is given by $$P(X=n, Y=m) =\frac{\alpha ^{m}\beta ^{n-m}}{m!(n-m)!}e^{-(\alpha +\beta )} $$ $$m=0,1,2...$$ $$n=m, m+1, m+2, ...$$ a) Evaluate $P(X=n)$ and $P(Y=m)$. b) Are $X$ and $Y$…
beepboopbeepboop
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