All Questions
Tagged with density-function random-variable
128 questions
2
votes
1
answer
34
views
What are some methods to estimate analytical PDF of random variable from an intricate expression of random variables?
Suppose, I have a random variable $y$ distributed as t- distribution
($\mu$) and a random variable $x$ distributed as gamma distribution ($\alpha,\beta$), and a variable $\theta$ distributed as ...
- 189
0
votes
0
answers
50
views
Distribution of a product of random variables
I have two independent distributions $X$ and $Y$. $X$ is defined by the piecewise CDF
$$F_X(x) = \begin{cases}
F_X^1(x) & x \in (-\infty, a_1)\\
F_X^2(x) & x \in [a_1, a_2)\\
F_X^3(x) & x \...
- 1
0
votes
0
answers
23
views
Finding the set for random variable transformations
I'm reading through the book "All of Statistics", and in section 2.12, regarding Transformations of Several Random Variables, the author lists three steps for finding the transformation. I ...
- 101
1
vote
0
answers
65
views
Joint density of two functions of a uniformly distributed random variable
I'd like to work out $\operatorname{Cov}(\cos(2U), \cos(3U))$ where $U$ is uniformly distributed on $[0, \pi]$.
I believe this involves computing $\mathbb{E}[\cos(2U)\cos(3U)]$. If so, then I first ...
- 345
2
votes
2
answers
132
views
If $X$ is a random variable, why is the PDF of $X + X$ not the same as the PDF of $2X$?
Background:
According to Wikipedia, the PDF of the sum of two random variables $X$ and $Y$ is given by the convolution:
$$f_{X + Y}(x) = \int_{-\infty}^{\infty} f_X(\eta) f_Y(x - \eta) \; d\eta$$
...
- 21
5
votes
1
answer
608
views
What is the resulting distribution if I merge two different distributions?
The title is not the best but I really do not know how to describe the scenario in a better way.
The context
Consider taking measurements of two different quantities:
The time needed for a car to ...
- 219
2
votes
0
answers
98
views
PDF of the sine of a wrapped Normal distribution
I have a random variable which is an angle $\Theta$ that follows a wrapped Normal distribution. The angle $\Theta$ has a relatively small variance, so despite having a range from $(-\pi,\pi)$, ...
- 131
0
votes
1
answer
85
views
Squared norm of linear system proportional to Multivariate Gaussian log-density?
I am reading https://epubs.siam.org/doi/10.1137/140964023, and I got confused by this part:
In the above, it is assumed that $m \geq n$.
If $m = n$, I can see how the above works.
$|| J \theta - y||^...
- 31
1
vote
1
answer
63
views
How to combine two integrals containing the PDFs of a variable and its linear transform?
Original Post:
Suppose we have two random variables $X$ and $Y$ with cumulative distribution functions $F(x)$ and $G(y)$. We know that $Y = aX + b$.
I want to compute $Z(x) = F(x) - G(y)$.
What I have ...
- 13
4
votes
2
answers
659
views
Operations on Random Variables vs Distributions vs Random Samples
What is the difference between i) random variables, ii) distributions of random variables, and iii) random samples?
While trying to figure out how to average random samples from various random ...
- 2,051
0
votes
0
answers
38
views
Find Marginal CDF probability from PDF (2 random variables) [duplicate]
Given the following PDF of continuous 2 random variables:
$$
f_{X,Y}(x,y)=\begin{cases}
y^2 & 0\le y\le x\le 1;\newline
0 & \text{otherwise}.
\end{cases}
$$
Graph showing the ...
- 217
2
votes
1
answer
560
views
Finding PDF/CDF of a function g(x) as a continuous random variable
Suppose that $X$ is a continuous random variable with PDF:
$$f_X(x)=\begin{cases}
4x^3 & 0<x\le 1\\
0 & otherwise
\end{cases}$$
and let $Y=\frac{1}{X}$. Find $f_Y(y).$
My approach:
First, ...
- 217
0
votes
1
answer
104
views
Cosine sequence with phase as random variable [closed]
I have a question on how to derive the (marginal) pdf for the $k$-th random variable $X_k$, where $X_k = \cos(0.2\pi k + \theta)$, and $\theta$ is a random variable uniformly distributed over the ...
- 1
2
votes
1
answer
103
views
The PDF of the random variable Z=Y+Y*X
I have two independent random variables, $Y$ and $X,$ where
$Y$ is a random variable with a Gaussian distribution and a zero mean.
$X$ is a random variable with a Gaussian distribution and a zero ...
- 21
2
votes
1
answer
203
views
Can CDF of a real random variable be a complex function? What does it mean physically?
I have a random variable $X$ which follows the following probability density function,
$$ p(x) = \frac{1}{4\pi} \Big[ \operatorname{erf}\Big(\frac{k\mu-x+2\pi}{\sqrt{2}k\sigma}\Big) - \operatorname{...
- 315
1
vote
0
answers
22
views
Esitmate of minimal of a function changed after transforming the variable
I want to perform MCMC or HMC for solving minimization problem of a function $f(x)$, then define the corresponding density $$g(x) = \exp\left(-f(x)\right)$$
Because the function of the future apply is ...
- 11
0
votes
0
answers
40
views
Computing Gini coefficient for a 2 parameters density function
I have a random variable $X$ defined by the following the density function,
\begin{equation}
f_{\theta_1, \theta_2}(x) =
\begin{cases}
\frac{\theta_1 \theta_2^{\theta_1}}{x^{\theta_1 + 1}}, &...
- 111
0
votes
0
answers
27
views
What is the pdf of the multiplication of two normal random variables? [duplicate]
I want to know the pdf of the multiplication of two normal random variables (may or may not have the same mu and sigma, may or may not be correlated).
...
- 542
7
votes
1
answer
839
views
How to find the PMF of a weighted sum of IID Bernoulli random variables with constant sum of weights
Let $\{X_1,X_2,\ldots X_k\}$ denote a set of $k$ IID $Bern(p)$ random variables. Also,
I have a set of $k$ non-negative integer weights denoted by $\{a_1,a_2,\ldots a_k\}$ such that $\sum_i {a_i}=k$.
...
- 224
1
vote
1
answer
70
views
Distribution Function from Density Function
I'm guessing there was an error in a Probability and Statistics exam I have recently taken.
Let $X$ be a random continuous variable and $f$ a function defined as follows:
$
f(x)=\left\{\begin{matrix}
...
- 13
3
votes
1
answer
425
views
Conditional probability mass function of number of Poisson random variable given their sum values
We have a discrete random variable $N$, and $X_1, X_2, ... X_N$ are i.i.d Poisson random variables with parameter $\lambda$. Denote $Y = \sum_{i=1}^{N} X_i$. What I want to know is:
If finding the ...
- 33
6
votes
2
answers
2k
views
How to write a joint kernel density of two random variables with known individual densities?
Consider two random variables $X$ and $Y$ with densities
$${f}_1(x) = \frac{1}{n_1h_1} \sum\limits_{i=1}^{n_1}K\left(\frac{x-u_i}{h_1}\right) ~~~~\text{and} ~~~~ {f}_2(y) = \frac{1}{n_2h_2} \sum\...
- 765
0
votes
0
answers
443
views
Sum of a number of shifted exponentially distributed random variables
I know that the sum of $k$ independent exponentially distributed random variables each with density function:
$$\displaystyle \lambda\,{{\rm e}^{-\lambda\,x}}$$
has an Erlang distribution:
$$\...
- 305
2
votes
1
answer
592
views
Change of variables in MCMC posterior
I have a question similar to this previously asked stackexchange question. However, instead of the expectation value after a change of variables, I am looking for the posterior probability density ...
- 123
1
vote
0
answers
41
views
Generating random variates knowing the density function
Let's consider a random variable that is following the distribution with the density function as below:
$$f(x) = \begin{cases} \sum_{i=1}^{\infty} 3.5i(0.3)^{i-1}e^{-5ix} & \text{for $x>0$} \\ ...
- 604
0
votes
0
answers
14
views
Discrete Random Variable Confusion
This seems like a simple question, but I am unsure. Please bear with me and thanks for the help.
I am told to suppose that A,B are discrete random variables that have a joint pdf, and am told to ...
- 101
0
votes
1
answer
510
views
finding probability distribution of sum of 2 random variables
I have a probabiliy distribution $$p(x) = \begin{cases}e^{-x} & x\geq0\\ 0 & x<0\end{cases}$$ I need to find the probability distribution for $Z=X+Y$ where X and Y are from the above ...
- 19
7
votes
4
answers
889
views
how to generate data from cdf which is not in closed form?
i am working on a distribution whose pdf and cdf is
$$f(x,\alpha,\beta)=\frac{(\frac{\beta}{\alpha})(\frac{x}{\alpha})^{\beta}}{(1+(\frac{x}{\alpha})^{\beta})^{2}}\frac{\sin(\frac{\pi}{\beta})}{(\frac{...
3
votes
1
answer
57
views
PDF of $X^2+2aXY+bY^2$
It is my first post on this forum. I am not a mathematician (so excuse me if I don't use the right vocabulary). I have two independent Normal random variables $X$ and $Y$:
\begin{aligned}
X&\sim N(...
-1
votes
0
answers
24
views
How can $X$ be a discrete random variable? [duplicate]
Suppose that the cumulative distribution function of discrete random variable $X$ is given by,
$$F(x) =
\begin{cases}
0 & \text{$x$ < 0 } \\[1.5ex]
\dfrac{x}{4} & \text{$0 \leq x<1$}\\[...
- 1
0
votes
0
answers
227
views
joint PDF of continuous and discrete random variables
Given exponential a random distribution X with PDF $f_X(x)=\lambda e^{-\lambda x}$ and a random variable Z with the PMF $p_Z[z]=0.5, z= \pm1$, I am trying to find the PDF of $Y=ZX$ (I also know that Z ...
- 1
0
votes
1
answer
363
views
How does a pdf change after a variable transformation with another random variable?
I have a probability density function of the energy $f(E)$ of a distribution of particles. Now, each energy gets shifted according to an angle $\theta$: $$E_{after} = E_{before} + g(E_{before}) \cos \...
2
votes
1
answer
2k
views
Random variable without pdf but with a cdf?
In this video, Blitzstein says that some random variables have no pdf but do have a cdf. Also, in my course material, I studied that converging in mean was stronger than converging in cdf which itself ...
- 462
1
vote
0
answers
48
views
Where does $X_n$ converge to?
Let $ X_1, X_2, X_3, \ldots $ be independent random variables and let $ X_n $ have a probability density fucntion (PDF) defined by
$ f_{X_n}(x) \quad=\quad \frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\...
- 11
2
votes
1
answer
663
views
Prove that $Z = \frac{X_1}{X_2}$, has an F-distribution
Let $X_1, X_2$ be independent random variables following density law $f(x) = e^{-x} , 0 < x < \infty$, Show that
$Z = \frac{X_1}{X_2}$, has an F-distribution.
I thought of solving this by ...
1
vote
1
answer
94
views
Does $\hat \theta_n=\theta+O_p\bigg(\dfrac{1}{\sqrt{n}}\bigg)$ imply that $p_{\hat \theta_n X}(x)=p_{\theta X}(x)+O_p\bigg(\dfrac{1}{\sqrt{n}}\bigg)$?
Let $X$ be a random variable. Let $\theta$ be a constant, and let $\hat \theta_n$ be a set of normally distributed random variables that converge in probability to $\theta$, that is, $\hat \theta_n \...
- 940
4
votes
1
answer
295
views
Probability Measure for Continuous Random Variable
Say a continuous random variable $X \in \mathbb{R}$ has a probability density function (p.d.f.).
Is it correct that,
change the probability measure $\Rightarrow$ define a new p.d.f. for X
?
In other ...
- 842
3
votes
2
answers
208
views
Generating random variates from the following pdf
I'm working through some example of probability distributions and I'm struggling to derive the formula for the following pdf
$f(x) = \frac{1}{0.02}e^{-\left\lvert x \right\rvert/0.01}$
My undersanding ...
- 61
1
vote
3
answers
2k
views
Is heights of humans actually a discrete random variable? [duplicate]
Suppose the human population consisted of $N = 3$ people, each with a specific height. Let $X^N$ be the random variable representing the heights of this population of $N$ people. Since $X^N$ can only ...
- 805
1
vote
1
answer
224
views
PMF of $aX_1 + bX_2$ (Bernoulli)
Let $Y_1 = aX_1 \sim \text{Bernoulli}(p)$ and $Y_2 = bX_2 \sim \text{Bernoulli}(p)$, what is the PMF of $Z = Y_1 + Y_2$ for $a > 0$, $b > 0$ and $a \neq b$?
Can somebody check my result?
$$p_{...
- 538
0
votes
0
answers
129
views
Bounding the norm of the difference between two related probability densities
Suppose we have a continuous random variable $X$ and two continuous functions $f$ and $g$ such that $f(X)$ and $g(X)$ are continuous random variables. Let $p_A$ be the probability density function of ...
- 893
2
votes
1
answer
161
views
Sum of probability density functions - can I treat this as a geometric series?
I have a random variable, $S_i$, that arises as the infinite weighted sum of another random variable $X_i$ in the form:
\begin{equation}
S_i = aX_i + a^2 X_{i-1} + a^3 X_{i-2} \ldots a^{n-1}X_{i-n+1}
\...
- 704
1
vote
4
answers
663
views
Probability function for difference between two i.i.d. Exponential r.v.s
My answer is completely off. Can you please tell me where did my logic go wrong.
Donald Trump and Tori Black are to meet at a specific time and both will be late by $ \sim Exponential(\lambda), i.i.d. ...
- 215
0
votes
0
answers
58
views
Are all these double integrals of the probability distribution of two continuous random variables equal?
If $X$ and $Y$ are two continuous random variables and $A$ and $B$ are any set within the range of $X$ and $Y$ respectively, are all these equal?:
\begin{align*}
P(X \in A, Y \in B)
&=\int_{X \in ...
- 555
2
votes
2
answers
285
views
Probability density function after transformation
Let $X,Z$ be random variables with probability density functions $p_X,p_Z$. Suppose $Z=f(X)$, where $f$ is continuous and differentiable. How is $p_Z$ related to $p_X$? It's tempting to say $p_Z(z) ...
- 6,738
1
vote
1
answer
759
views
Is pX(Y) a random variable or a number?
I reason that is a random variable because Y is a random variable, thus making Px acting randomly. Example Y sample space is a roll of a die (1,2,3,4,5,6). So any of those values could be inputed in ...
- 257
0
votes
1
answer
448
views
Transformation of a random variable with a gamma distribution
Suppose $X_i \stackrel{i.i.d}{\sim}$ Exp$(1/\theta)$ which implies $\sum_{i =1}^{n} X_i \sim$ Gamma $(n, 1/\theta)$.
But, then, the book that I am reading says that $(2/\theta)\sum_{i =1}^{n} X_i \...
- 239
1
vote
1
answer
52
views
Why condition on either the r.v. $X$ or $Y$ and integrate over a product of pdfs rather a single pdf to find this probability density?
Let $X$ have the probability density $f_{X}(x)=\lambda e^{-\lambda x},
\;\; x>0$ and let $Y$ have the probability density $f_{Y}(y)=\lambda
e^{-\lambda x},\;\; y>0.$ Find the probability ...
user271077
1
vote
1
answer
742
views
Find joint pdf table of two discrete independent random variables $X$ and $Y$
Given the pdfs of two discrete independent variables $X$ and $Y$, write the joint pdf. There is a property that $ if\ \ p_{XY}(x,y) = p_X(x)p_Y(y) \ \forall i,j
\Rightarrow \text{X,Y are ...
2
votes
1
answer
1k
views
example of when the likelihood function does not sum up, or integrate to $1$? [duplicate]
Could someone please give an example of when the likelihood function does not sum up, or integrate to $1$? I have seen this question with the first answer but it really confused me - why are we ...
user255658