Questions tagged [pdf]

Probability density function (PDF) of a continuous random variable gives the relative probability for each of its possible values. Use this tag for discrete probability mass functions (PMFs) too.

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3
votes
1answer
74 views

Change of variables in pdf

I have the joint pdf$$f(x_1,x_2)=x_1e^{-x_1(1+x_2)}I_{(0,\infty)}(x_1)I_{(0,\infty)}(x_2)$$and have to derive the joint pdf of $$Y_1=e^{-X_1}\qquad\text{ and }\quad Y_2=e^{-X_1X_2}$$ I set $x_1=-\ln(...
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0answers
13 views

Differentiation of Erlang Cumulative Distribution Function (CDF) analytically [migrated]

I have an idea of Erlang PDF by intuition but I want to get the answer by analytical derivation of its CDF i.e. \begin{equation} F_{Y_k}(y)=1-\sum_{n=0}^{k-1}\dfrac{(\lambda y)^n e^{-\lambda y}}{n!} \...
0
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0answers
30 views

Sum of two iid random variables

Let $X_1$ and $X_2$ be two iid random variables with pdfs $f(x_i)=e^{-x_i}I_{(0,\infty)}(x_i)$. How can I determine the pdf's of $S=X_1+X_2$ and $D=X_1-X_2$? I know how to invert a pdf when there are ...
3
votes
1answer
147 views

Excepted conditional density and conditional expectation

Apparently one can obtain a regression analysis as $$g(x)=\frac{\int yf(y,x)dy}{f(x)}$$ where $$f(x)=\int f(y,x)dy$$ is the marginal density of $X_i$. In effect, I believe, the above expression ...
39
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10answers
40k views

Why is the sum of two random variables a convolution?

For long time I did not understand why the "sum" of two random variables is their convolution, whereas a mixture density function sum of $f(x)$ and $g(x)$ is $p\,f(x)+(1-p)g(x)$; the arithmetic sum ...
0
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1answer
17 views

distribution of data vs density estimation

I think this is a very basic question but I'm confused and need clarification please help me density estimation vs distribution of data what is the difference and Is Kernel density estimation learns ...
0
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0answers
17 views

How can I interpret a kernel density plot where x axis is the probability?

I am confused on how to interpret the kernel density estimation (kde) plot given below, which is of the predicted probabilities from my model for class 0 and class 1. What conclusions can I draw from ...
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0answers
51 views

Finding PDF of max of sum of truncated sinusoids

Suppose I am adding N sinusoids. The frequencies, phases, and amplitudes are chosen to be iid in the range f1 to f2, 0 to 2pi, and A1 to A2 respectively. Now, if I am observing the sum from time t1 to ...
-1
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0answers
35 views

cdf $F_{\theta}(x) = 1 - (\frac {\theta_1}{x})^{\theta_2}$ for $\theta_1 < x$ then find MLE of $\theta_1, \theta_2$ [closed]

Question: $X_1,\dots,X_n$ iid observation from cdf $$F_{\theta}(x) = \begin{cases} 1 - (\frac {\theta_1}{x})^{\theta_2} & \theta_1 < x \\ 0 & \text{otherwise} \end{cases}$$ Find MLE of $\...
1
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0answers
31 views

Find $x_0$ that satifies $\mathbb{P}(X \leq x_0) = 0.75$

Suppose that $X$ is a continuous random variable with probability density function: $$\begin{cases} x & 0 \leq x < 1, \\[6pt] 2-x & 1 \leq x < 2, \\[6pt] 0 & \text{otherwise}. \\...
0
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1answer
36 views

Probability density functions

The probability density function of $X$ is defined by: \begin{align} f(x) = \begin{cases} \alpha & \quad, 0 \le x \le 1 \\ \beta(x-4)^2 & \quad, 1 \le x \le 4 \end{cases} \end{align} ...
0
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0answers
20 views

Simulating p-value statistics for Liliefors test (Python statsmodels)

Python statsmodels has an implementation of Lilliefors' test for goodness of fit (i.e. if the parameters of the distribution were obtained from fitting the data and not per-determined as in the ...
4
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1answer
59 views

Can this statistic be shown not to be sufficient for $\theta$?

This problem comes from Casella and Berger, who do not rigorously demonstrate (in their solution key) that the statistic is not sufficient. Let $X_1,\dots,X_n$ be a random sample from a population ...
1
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1answer
28 views

How to Evaluate a Probability Density Function you Fitted?

I was reading "A Student's Guide to Bayesian Statistics" by Ben Lambert and he brought up something I never thought of before and can't find the answer to on Google. That is, evaluating a ...
1
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1answer
38 views

3-part question on joint PDFs

a.) Let U, V be uniformly distributed over the set $\{(u,v): $$0<u<v<1$}. Let $X$ = $-$$log(U)$, $Y$ = $-$$log(V)$, $Z$ = $max$($X$,$Y$). a.) Draw the support of the joint distribution ($U$, $...
3
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1answer
52 views

Is it possible for a distribution to have a non-zero probability of generating a value with zero probability density?

I am reading the book "An Elementary Introduction to Statistical Learning Theory" and there is a sketch of a proof (Section 8.4) for the universal consistency of kernel rules for binary ...
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0answers
13 views

How to estimate parameters and pdf of a random variable transformed from a lognormal random variable?

I have a continuous random variable Y that follows lognormal distribution with known parameters (mu and sigma). Let Y be transformed to X=Y-20000. So it is basically shifted to left. How do I find the ...
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0answers
10 views

Finding PMF, CDF of a piecewise function of an RV

Here's the question: Let $Z$ have CDF $F$ and pdf $f$ and let $A$ be a subset of the real line. Further, let \begin{cases} W = 1 & \text{if $Z \in A$} \\ ...
3
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0answers
102 views

Does this distribution have a name? $p(x) \propto |x|^a \exp\left(-\frac{1}{2} (x-b)^2 \right)$

Quick question. Anyone able to attribute the following kernel to a known probability distribution (univariate, continous on the real line)? $$ p(x) \propto |x|^a \exp\left(-\frac{1}{2} (x-b)^2 \...
0
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0answers
23 views

Expectation (x/y) using jacobian

Let $X$ and $Y$ be two independent random variables with the density functions: $f(x) = 3 x^2$, for $0<x<1$, $0$ elsewhere $g(y) = 4y^3$, for $0 <y<1$, $0$ elsewhere Give $\mathbb E(x/...
1
vote
1answer
55 views

finding PDF of Y, given Y|X [closed]

$$Y|X\sim Bin(X,n)$$ $$X\sim U([0,1])$$ How can I find the PDF of Y? I know that: $$\Bbb P(Y=k)=E_X[\Bbb P(Y=k)|X]$$
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0answers
26 views

How can we perform the integration for showing this equation?

Im reading this paper momentarily, and there is one equation (9) in section 3.1. that I just can't wrap my head around yet: \begin{align} \mathcal{N}(\textbf{y}_d;\textbf{0},\pmb{\Phi \Phi}^T + \...
0
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1answer
20 views

Calculate the likelihood from the density function with known mean and sd

In this link the likelihood of IQ has been calculated by using dnorm function in R. Here they used "%" sign but based on the range of ...
1
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1answer
36 views

Density of square root of sum of squared independent uniform random variables [duplicate]

Let $X \sim U(-1, 1)$ and $X \sim U(-1,1)$. We want to find density function of $W = \sqrt{X^2 + Y^2}$. I got stuck and I have no idea, where I am making a mistake. This is my approach. Let $F$ be a ...
0
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1answer
167 views

Visualize Covariance when only probability mass and marginal functions are given

I am trying to intuitively understand Covariance like here. So if a random sample set given, I could draw rectangles with them, one of the cornes being fixated on mean $(\overline{x},\overline{y})$. ...
0
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1answer
104 views

CDF for $f(x) = 0.5e^-|x|$

This is the full question: "If a random variable has density $f(x)= 0.5e^{-|x|}$, for $x\in R$, find the cumulative distribution function". I know that to find cdf from the pdf you would ...
1
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2answers
41 views

How do I find the PDF from a multidimensional CDF with indicator functions?

I have what I'm sure is a very stupid question. When I have a two-dimensional random variable $\tilde{X}=(X_1,X_2)$ with the cdf $F(x_1,x_2)=(kx_1^2I_{(0,1)}(x_1)+I_{[1,\infty)}(x_1))(kx_2^2I_{(0,1)}(...
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0answers
9 views

What are some examples of multiparameter probability distributions with three or more parameters?

What are some examples of multiparameter probability distributions with three or more parameters? What real life phenomena are they used for modelling?
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0answers
36 views

How do I show this equation involving the Gaussian pdf? [duplicate]

Im reading this paper momentarily, and there is one equation (9) in section 3.1. that I just can't wrap my head around yet: \begin{align} \mathcal{N}(\textbf{y}_d;\textbf{0},\pmb{\Phi \Phi}^T + \...
1
vote
1answer
278 views

Mean and variance of maximum of normal random variables

I'm trying to find the mean and variance of $Y = \max(X_1, ..., X_n)$ where $X_i \sim \mathcal{N}(\mu_i, \sigma^2)$. Note that the $X_i$ are independent, but not identically distributed. That is, ...
1
vote
1answer
79 views

Conditional probability density from probabilities

I am trying to understand conditional probablility densities in relation to the conditional probablilities. From the Measure-theoretic definition on Wikipedia, if $X$ and $Y$ are non-degenerate and ...
4
votes
1answer
43 views

How to get a PDF which converts an already drawn sample to uniform [closed]

Suppose i have a large data pool with a particular PDF, $F(x)$, interval $[x,y]$ estimated from KDE of the datapool. I drew $N$ samples at random from that data pool and saw that their distribution is ...
1
vote
1answer
107 views

Given a multi-dimensional sample, how do I build a distribution density coefficient?

Given a sample $X=\{\vec{x}_1, \dots,\vec{x}_l\}$ where $\vec{x}_i \in \mathbf{R}^d$ with $d>3$: I would like to know if it's possible to have and index that is inversely proportional to the ...
1
vote
1answer
36 views

Marginal distribution

A loss distribution has PDF - $f(x) = 1/x^2$, for $x > 1$ An insurer finds that the time in hours it takes to process a loss amount x has a uniform distribution on the interval $(\sqrt x, 2\sqrt x)$...
0
votes
1answer
44 views

An example of continuous random variable X > 0 with finite second moment but Infinite third moment [duplicate]

Can someone construct an example of this? i.e., $E[X^2] < \infty$ but $E[X^3] = \infty$. Results could be in terms of pdf, or cdf, or survival function. Justification would be appreciated
3
votes
1answer
25 views

Finding P(a< u(X,Y) <b) given a rectangular support

>The continuous variables X and Y have the following joint pdf $f(x,y) = x + y, 0<x,y<1.$ Determine $P(0.5<X+Y<1.5)$. I know that the support of x and y is rectangular, hence they are ...
5
votes
2answers
286 views

What does $N(x|\mu, \sigma^2)$ mean?

I am supposed to show that $f(x) = \sum_{k=1}^{K}\pi_k N(x|\mu_k, \sigma_{k}^2)$ complies with the properties of a density function but I have no idea how to do this since I am not sure what $N(x|\...
2
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1answer
33 views

How do you work with a function of a uniform distribution? [closed]

I am struggling with parts b and c. How do you solve them? Could you please give the solution?
0
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0answers
13 views

evaluating an empirical multivariate PDF in python

I have multivariate (bivariate in the simplest case) residuals from a VAR time series regression and I'd like to estimate the joint pdf and then be able to draw from this pdf. If I have bivariate ...
1
vote
1answer
24 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 ...
0
votes
1answer
27 views

Why does Uniform distribution make sense?

This might be a dumb question, but I am suddenly confused on how to understand the PDF of a uniform distribution. For instance, the PDF of standard uniform is always equal to 1... How is that ...
0
votes
0answers
24 views

Showing a minimal sufficient statistic [duplicate]

If we have common density $$f(x|\theta)=\theta^{-1}x^{\frac{1-\theta}{\theta}},$$ with $x\in(0,1)$, $\theta>0$ and $\textbf{X}=(X_1,...,X_n)$ is a random sample. Then how can we show that the ...
3
votes
1answer
38 views

Finding C for which f(x) is a density function

One of the points of the exercise states: Find the constant $C$ for which the following function is a density function $$ f(x)= \begin{cases} C(x-x^2) & 0 \leq x \leq 2\\ 0 ...
0
votes
1answer
25 views

Why are there two ways to write PDF and CDF functions?

I often see PDF and CDF functions written as either $f_X(x)$ or $f(x)$ for PDF or $F_X(x)$ or $F(x)$ for CDF. In what situations would you use either notation? Like what is the point of having ...
0
votes
1answer
37 views

Finding the pdf of $X_{(1)}$ of the two-parameter exponential distribution

I have to find the pdf of the smallest order statistic $X_{(1)}$ of two-parameter exponential distribution whose pdf is: $f(x; \theta_1, \theta_2) = \frac{1}{\theta_2} \exp\{-\frac{x-\theta_1}{\...
0
votes
1answer
32 views

The joint density of two dependent —yet uncorrelated— normal variables

Consider this setup. Let $X$ be a standard normal variable. Let $I$ be a Bernoulli random variable with parameter $p= 0.5$. Let $Y$ be equal to $(2I - 1)\cdot X$. In the link I provided it is shown ...
0
votes
1answer
49 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 \...
65
votes
4answers
30k views

What is the reason that a likelihood function is not a pdf?

What is the reason that a likelihood function is not a pdf (probability density function)?
7
votes
1answer
156 views

Finite state machine with gamma distributed waiting times

I have a state machine with positive and negative inputs. The time between positive inputs follows a gamma distribution ($X_+ \sim \Gamma(k_+, \theta_+)$), and the time between negative inputs follows ...
0
votes
0answers
23 views

Censored data - When does it matter

In survival analysis, one may arrive at a series of samples $X_1,...,X_n$, for which the outcome of a given $X$ may not be "observed" within the experiment. For instance, if the $X_i$'s are failure ...

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