All Questions
Tagged with density-function expected-value
47 questions
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Expected Value for Complex-Valued Random Variable
This question is part of Exercise 3.14 in the book The Analysis of Time Series: An Introduction with R, 7th Ed., by Chatfield and Xing.
Problem Statement: Suppose $\theta$ is uniformly distributed ...
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0
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25
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Question on the proof step in the theorem 1 of the Gap statistic paper
From the Gap statistic paper, during the proof for the theorem 1, we can see the below equality (p. 422),
$\begin{aligned} \operatorname{var}(X) & =\frac{1}{2} \int_{-\infty}^{\infty} \int_{-\...
- 303
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0
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65
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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 ...
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54
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Beginner Probability Question: Find PDF and E(X) [closed]
From [1;2] continuous interval we choose 3 numbers randomly. Let $X$ be the minimum between those numbers.
Find PDF and Expected value.
I fail to understand the problem, since I believe that ...
1
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2
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205
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Calculate mean of density function (example using lognormal distribution)
In a lognormal distribution, the mean is equal to $\exp(\mu + \frac{\sigma^2}{2})$.
I tried to separately calculate this using the definition $E[X] = \int{xf(x)dx}$, where I have 200 $x$ values evenly ...
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1
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115
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How do we define the pdf in the multi-variate case and compute expectations?
Apologies if this is a very simple question but trying to work through a result in a paper made me realize I missed something a bit fundamental in my undergrad probability and analysis courses. Lets ...
- 115
3
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1
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177
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Calculate expected values E(x) & E(y) & variance of x & y of joint PDF, which was previously transformed from Polar to Cartesian
Given two independently uniform distributed random variables angle $\theta \in [0,2\pi]$ and radius $r \in [0,1]$.
I obtain for the joint density function with polar coordinates: $$ f_{r,\theta}(r,\...
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0
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51
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Derive E[Y|X] when the joint probability is given
Now, consider joint density of $X, Y$ :
$$
f_{X, Y}(x, y)=\left\{\begin{array}{l}
\frac{1}{\pi} ; X^2+Y^2<1 \\
0 ; \text { Otherwise }
\end{array}\right.
$$
Derive $E(Y \mid X)$.
I know how to ...
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7
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1
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447
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Continuity of the location
Consider $f(x,\theta),$ a density function for the random variable $X$ with a parameter $\theta$. Suppose $f(x, \theta)$ is continuous in $\theta$.
Is the location
$$\int_\mathbb{R} x f(x, \theta) dx$$...
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1
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1
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645
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Expected Value of Exponential CDF
I am given the following CDF and I want to calculate its expected value:
$F(Y \leq y) =1-( 0.28e^{-0.5y} + 0.71e^{-0.25y})$
Creating the PDF:
$f(Y \leq y) = \frac{71\mathrm{e}^{-\frac{x}{4}}+56\mathrm{...
- 13
0
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0
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25
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Expected value of X given Y is less than some constant [duplicate]
Here is the problem I'm trying to work out: Let $v_b,v_s$ be jointly normally distributed random variables with pdf $f(v_b,v_s)$. I want to work out $E[v_b|v_s\leq\pi]$ for some constant $\pi$. Here ...
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1
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70
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Expectation of ratios of probability density functions
I'm trying to solve/simply the expression below:-
$\large \mathbb{E_{x \sim b(x)}} B\ [log\left(1 - \frac{A\ a(x)}{2\ c(x)}\right)]$,
or
$B \large \int_{x}b(x)log\left(1 - \frac{A\ a(x)}{2\ c(x)}\...
- 33
1
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1
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90
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Interpretation of $xf_X(x)$, the integrand from the expected value calculation
Seeking the expected value of a continuous random variable we calculate the integral $\int_{-\infty}^\infty xf_X(x)\ dx$.
Does the integrand $xf_X(x)$, i.e. the product of $x$ and the corresponding ...
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1
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286
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notation in expectation of continuous random variable question
What does the 'x' over the '2' mean in the statement of the values this function takes
- 283
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1
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443
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Expectation of inverse square under multivariate standard normal
In one of the steps in my lecture notes, the following result was used without proof:
Given $X$ is a $p$-dimensional multivariate normal distribution, where $p\ge 3$, centred on zero, with covariance ...
2
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1
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2k
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Expected Value for 2 Random Variables with Joint Probability Distribution
I have trouble with determining the domain for integration in the case of having a joint pdf when one variable depends on the other. There are two examples I don't quite understand, and hopefully, you ...
- 175
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2
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3k
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Expectation with respect to a transformed random variable
Problem
Suppose I have a random variable $z$ following a distribution $p(z)$. Suppose I have a transformation
$$
f(z) = x
$$
that transforms the random variable $z$ into a new random variable $x$ with ...
- 2,286
0
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0
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271
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Expectation (x/y) using jacobian
Let $X$ and $Y$ be two independent random variables with the density
functions:
$f(x) = 3 x^2$, for $0<x<1$, $0$ elsewhere
$g(y) = 4y^3$, for $0 <y<1$, $0$ elsewhere Give $\mathbb E(x/...
1
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1
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46
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Marginal distribution
A loss distribution has PDF - $f(x) = 1/x^2$, for $x > 1$
An insurer finds that the time in hours it takes to process a loss amount x has a uniform distribution on the interval $(\sqrt x, 2\sqrt x)$...
0
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41
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Relating $E[X]$ and $E[h(X)]$ by 'adjusting' the PDF
I have just started studying statistics and I have been introduced to PDFs and expected values. Now the formal definition of $E[X]$ for a continuous random variable is $\int_{\mathbb{R}}tf(t)\mathrm ...
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1
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389
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Find expected value with pdf and LOTUS
I am currently trying to solve a problem and can't figure it out. I have done this before, but I can't remember all of the details and can't find a reference example.
Let's say I have a pdf
$$f(x)=\...
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1
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248
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How do you calculate the expectation without a pdf in the context of Center limit theorem, variance
Given Problem 1)
To get the variance and covariance the following steps are taken:
In the step below to calculate E[Z^2] how do we approach this without a known pdf? For completeness would finding E[...
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-1
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1
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331
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Correct or Not? Probability and Geometry [closed]
A stick of length $1$ is broken into two pieces of length $Y$ and $1−Y$ respectively, where $Y$ is uniformly distributed on $[0,1]$. Let $R$ be the ratio of the length of the shorter to the ...
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4
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4k
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How to make sense out of integration over discrete data points?
Looking for a proof of the expected value of the score function equating zero, I came to this document that was recommended in another answer.
Considering that we have a sample of $n$ $x_i$ values, I ...
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1
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4k
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Notation: What does the tilde below of the expectation mean? [duplicate]
I am reading about variational auto encoders, and there is the below loss function:
$$l_i(\Theta,\phi) = - {\mathbb{E}}_{z\sim q} \left[\log p_\phi(x_i|z)\right] + KL(q_{\phi}(z_i|x)||p(z))$$
What ...
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1
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What is the expected value of half a standard normal distribution?
You have a normal distribution with mean of 0 and variance of 1. Keeping the same probabilities and focusing only on half of the distribution (other half has it's original probabilities but x values ...
- 3,263
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1
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337
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PMF and independence with two discrete random variables?
Each of n people (whom we label 1, 2, . . . , n) are randomly and independently assigned a number from the set {1, 2, 3, . . . , 365} according to the uniform distribution. We will call this number ...
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1
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290
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Expectation of $h \circ X$
I'm only starting to learn statistics.
The definition I've been given for the expected value (expectation) of a continuous random variable X with probability density function (PDF) $f_X$ is the ...
2
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1
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35
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Mean nonequality test if data are intervals
If $X, Y$ are two sets of observations of two random scalar (univariate) variables, one can determine if the expected values of the two variables are unequal, with appropriate tests. My question is: ...
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585
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Expectation of the inverse of $\textbf{z} \textbf{z}^{H}$ where $\textbf{z}$ is a complex Gaussian vector
Considering the vector $\textbf{z} \sim \mathcal{CN}(\textbf{0}_{M},\Theta_{M \times M})$, what would be the expectation of $\frac{1}{\textbf{z} \textbf{z}^{H}}$, i.e.,
$\mathbb{E} \left\lbrace \frac{...
-1
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1
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1k
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Getting negative variance
I'm having a problem when calculating the variance of the following estimator:
$\hat\theta=\frac{1}{N}\sum_{n=1}^{N}D_n$ with $D_1....D_N$ independent random variables.
In order to calculate the ...
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1
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3k
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Mean and Variance of the Area of a Circle with Uniform Radius
A circle with a random radius R∼Unif(0,1) is generated. Let A be its area.(a) Find the mean and variance of A, without first finding the CDF or PDF of A.(b) Find the CDF and PDF of A.
So, quite ...
2
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1
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1k
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Expectation using CDF
I have a problem understanding the solution of an exercise:
$F_x(x) = 1-(\frac{\sqrt{3}}{2}x + \frac{3}{2})^{-3}$ for $x \geq \frac{-1}{3} \sqrt{3}$, $0$ elsewhere
and i am asked to compute the ...
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Expected value with piecewise probability density function (PDF)
I am continuing the prepare for an exam by reviewing handouts from an old statistics course I took. The handout came with a set of solutions prepared by the instructor, but I suspect that one of the ...
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Expectation of a strictly increasing function
Assume that $X_1$ and $X_2$ are two i.i.d. random variables with pdf $f$. Also, assume that $a$ and $b$ are two fixed real numbers such that $a>b$. If $g$ is a strictly increasing function, do I ...
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1
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73
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Finding the expected value of a continuous random varibale when the commulative distribution is given
I have this distribution function of a random variable X:
I wish to find E(X).
I have used derivatives to get the density function, compared it to 1, and found that f(t) = (4/5)t+(3/5). I then used ...
- 815
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0
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833
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Joint Density and Covariance between Two Random Variables with the same Mean and Variance
This seems like a deceptively simple question, (and it perhaps is and I am missing something) but I could not find anything on this.
Q1)
Are there any general results / relationships to get the Joint ...
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478
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Compound Distributions --- Basic Techniques and Key General Results from First Principles
Could someone please point me to a source with notation, terminology, key results and basic techniques to approach compound distributions?
Definition
Compound probability distribution is the ...
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1
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72
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Help to understand this. Expected value of $S^\alpha$ in Gaussian distribution [closed]
Lets $X_1,\cdots,X_n$ be simple random sample from $\mathcal{N}(\mu,\sigma)$.
$\overline{x}$ is sample mean.
Let
$$S^2=\begin{cases}\sum_{i=1}^n (x_i-\mu)^2, \mathrm{ where\ } \mu \mathrm{\ is\ ...
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3
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How to calculate the expected value of a standard normal distribution?
I would like to learn how to calculate the expected value of a continuous random variable. It appears that the expected value is $$E[X] = \int_{-\infty}^{\infty} xf(x)\mathrm{d}x$$ where $f(x)$ is the ...
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62
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Risk in density estimation: grasping the definition
When generalizing estimators to an entire function what is the space in which we perform the integral to obtain the expected value (with respect to this function)?
For example, when estimating ...
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3
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2k
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Let f(x) be some PDF, and F(x) be its CDF. Shouldn't F(x)=.5 give us the expected value of f(x)?
I was playing around in R and have gotten myself very confused about the relation between probability distributions, their expected values, and their cumulative distribution functions.
Say we're ...
user44845
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386
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Expectation of density ratio of two iid variables
Let $X \sim N(0,1)$ and $Y \sim N(0,1)$ be independent RVs and let $f$ be their density function. I'd like to compute the expectation of the density ratio
\begin{align}
\mathbb{E}\left[\frac{f(X)}{f(Y)...
- 308
3
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2
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256
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What is the expected partial value function really called?
If f is a pdf, the integral of xf(x) over the entire range where f(x) > 0 gives, of course, the expected value. Suppose that integrate the same function, xf(x) from negative infinity up to t, ...
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120
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I want to prove that these definitions of expected value hold
Let $(\Omega,\mathcal B,P)$ be a probability space. I have two (related) questions. Assuming that $g:\mathbb{R}\to\mathbb{R}$ is Borel measurable, and understanding that
$$E(g(X)) = \int_{\Omega}g(X(...
- 403
9
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1
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3k
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What is the meaning of subscript in $p_{\theta}(x)$ and ${\mathbb E}_{\theta}\left[S(\theta)\right]$?
In the context of likelihood-based inference, I've seen some notation concerning the parameter(s) of interest which I've found a little confusing.
For example, notation such as $p_{\theta}(x)$ and ${...
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2
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3
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3k
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Expected value of a random variable in a range
I have a random variable $x$ with $E(x) = \mu$ and PDF $f(x)$ and CDF of $F(x)$.
Is there any way to represent the $E(x|x<\bar{x})$ in terms of $\mu$ and $f(x)$ or $F(x)$?
i.e. to write the ...
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