Questions tagged [expected-value]
The expected value of a random variable is a weighted average of all possible values a random variable can take on, with the weights equal to the probability of taking on that value.
2,272
questions
0
votes
0
answers
9
views
Is There a Standard Metric for Evaluating Treatment Impact Considering Action Cost in Uplift Models?
I'm currently exploring Uplift modeling, specifically the use of the Conditional Average Treatment Effect (CATE) metric:
$$ \tau(t', t, x) := \mathbb{E}[Y | X=x, T=t'] - \mathbb{E}[Y | X=x, T=t] $$
...
- 115
2
votes
1
answer
27
views
MCMC Intuition for Expected Value Estimate
I believe I have a fundamental misunderstanding of MCMC.
We seek to estimate the expected value of a target distribution using its integral form.
We consider a markov chain of samples from a simpler ...
- 33
1
vote
0
answers
73
views
Estimating expected value with respect to posterior
I have a neural network and I need to calculate the following:
$$\mathbb{E}_{P(\theta|D)}[f(\theta)]=\frac{\sum_\theta P(D|\theta)P(\theta)f(\theta)}{\sum_\theta P(D|\theta)P(\theta)}$$
Where $f$, ...
- 253
5
votes
1
answer
175
views
expectation value, distribution function and the central limit theorem
The problem goes thus:
${\{X_n\}}$ is an $iid$ sequence of random variables with mean 0 and variance $\sigma^2$. If the third moment is finite, show that $$\lim_{n \to \infty} \mathbb{E} \left(\left(...
- 53
3
votes
2
answers
92
views
Expected value after $K$ Bernoulli trials where the $i$-th probability of success depends on the current number of successes
I have an experiment that involves $K$ Bernoulli trials. Trial $i$ has probability of success $p_{i, n}$ where $n$ is the current number of successes (so $0 \leq n \leq i-1$).
If my random variable is ...
- 157
1
vote
1
answer
73
views
Expectation of the reciprocal of a standard normal random variable [duplicate]
If $\mathbf{X} \sim_{iid} \mathcal{N}(\mu, 1)$ then we know that the sample mean $\bar{X} \sim \mathcal{N}(\mu, 1/n)$, how would we show that $$\mathbf{E}\left(\frac{1}{\bar{X}}\right) = \infty $$ and ...
- 41
0
votes
0
answers
17
views
Taking expectation of higher moments using original distribution [duplicate]
If we want to calculate $\mathbb{E}[X]$, we know this is $\int_{-\infty}^{\infty} x f_X(x) dx$.
When we want to calculate $\mathbb{E}[X^n]$ (for, say, positive integer $n$), we also do this with ...
- 111
7
votes
2
answers
676
views
Is the expectation of a random vector multiplied by its transpose equal to the product of the expectation of the vector and that of the transpose
I'm taking a course in advance statistics and we have to prove whether the following expression is true: $E[zz^T]=E[z]E[z^T]$. I am assuming it is not, since the formula of the covariante matrix is $...
- 101
1
vote
1
answer
111
views
Show that for random variable $X$ with $N = \{1, 2, \ldots \}$, $E(X) = \sum_{n = 1}^\infty P(X \geq n)$ [duplicate]
Prove that for random variable with natural numbers from 1 to infinity the expected value $E(X)$ is equal to $\sum_{n = 1}^\infty P(X \geq n)$. Is this the mathematically correct way to prove it? And ...
- 35
2
votes
2
answers
84
views
Interview Question: What is the probability they will be home in more than 30 minutes?
The following is an interview question:
A student leaves Univeristy (U) to walk Home (H). It is a distance of 4 blocks in a straight line. At each crossing, they toss a coin deciding whether to move ...
- 381
0
votes
0
answers
21
views
How to prove that the MLE of a uniform distribution is biased using the formula given below? [duplicate]
I've calculated the MLE of the uniform distribution on [0,theta] as maxi{Xi} but don't know how to prove it is biased. The formula I have learned to prove it is unbiased is E(θ^)-θ=0.
Was stuck on how ...
- 1
2
votes
1
answer
100
views
Moments of sum of squares of independent gaussians $X_i \sim \mathcal{N}(\mu_i,\sigma^2_i)$, or $||X||^2$
Say that we have $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$. Is there some formula to calculate analytically the expected value of the sum $S = \sum_i^n X_i^2$?. This is equivalent to computing $\...
- 1,077
0
votes
0
answers
19
views
Find function $f$ such that the performance of models $h_1$ and $h_2$ is similar
Suppose that $h_1$ and $h_2$ are the first and third order polynomials respectively, which are obtained by solving the OLS equation (using the training dataset). Also consider the following ...
- 101
3
votes
1
answer
67
views
Given $r\gt 0$, how to get $\mu_r = E[|U|^r]$ where $U\sim N(0,1)$?
Given a standard normal random variable $U$ , is there a general formula to compute the expected value of the absolute value of $U$ to any power?
For example given a non negative constant $r$ (i.e $r\...
- 83
7
votes
2
answers
150
views
Marginalising out two variables from an expectation
Lets say I have the following expected value:
$$E[Y|X_1=x_1,X_2=x_2,X_3=x_3]$$
and I want to marginalize out the continuous random variables $X_2$ and $X_3$ to arrive at:
$$E[Y|X_1=x_1]$$
How can I ...
- 1,162
0
votes
1
answer
49
views
Beginner Probability Question: Find PDF and E(X) [closed]
From [1;2] continuous interval we choose 3 numbers randomly. Let $X$ be the minimum between those numbers.
Find PDF and Expected value.
I fail to understand the problem, since I believe that ...
0
votes
1
answer
48
views
Calculate $E[X]^2$ where $X \sim \operatorname{Binomial}(n,p)$ with binomial coefficients expansion [closed]
Calculation of $EX$ using the binomial expansion formula is easy:
\begin{align}
EX &= \sum_{x=0}^{n}x\frac{n!}{(n-x)!x!}p^{x}(1-p)^{n-x}\\& = np \sum_{x=1}^{n}\frac{(n-1)!}{(n-x)!(x-1)!}p^{x-...
- 11
0
votes
1
answer
31
views
Convergence of a normalized density weighted sum in the limit
Let $X \sim f_X$ and assume that for each $N \in \mathbb{N}$ we have datasets of $N$ iid samples $D_N = \{ X_i | 1 \leq i \leq N \}$. Also assume that there is some bounded deterministic function $g(...
- 3
3
votes
2
answers
143
views
Inequalities involving expectations
Consider four random variables $W, X,Q,Y$, where $Q$ and $Y$ are binary. Assume
$$
\begin{aligned}
& (1) \quad E(Q(X+W)|Y=1)\geq 0\\
& (2) \quad E(W|Y=1)=0\\
& (3) \quad \Pr(Q=1|Y=1,W)=1
\...
- 716
0
votes
0
answers
27
views
Why $\mathbb{E}_{(x, y) \sim \mathcal{D}}[f] = \mathbb{E}_{x \sim \mathcal{D}_{X}}[\mathbb{E}_{y \sim \mathcal{D}_{Y|x}}[f|X=x]]$ [duplicate]
I found this equality on p.6 in this document proving that Bayes Predictor is optimal (i.e. it achieves the minimal generalization risk) amongst al hypotheses:
$$
\mathbb{E}_{(x, y) \sim \mathcal{D}}[\...
- 478
0
votes
1
answer
82
views
"Almost surely" used in an expectation
Let $(\mathsf{X}, \mathcal{X})$ be a measurable space, $\pi(dx)$ be a probability measure on it, and $K:X\times\mathcal{X}\to[0, 1]$ be a Markov kernel. I have the following property
$$
\int K(x, A) \...
- 657
2
votes
0
answers
87
views
What is the expected length of an interval on an arc of a circle that can be constructed using exponential variates?
I had asked this question on Math stackexchange once before and now again but this does not seem to be drawing too much attention. Since this is a question that can be safely classified as non-measure ...
- 71
0
votes
0
answers
22
views
Calibration Expectation Decompostion
I am reading a "Calibrated Structured Prediction by Kuleshov and Liang" link.
Calibration and sharpness. Given a forecaster $F : X → [0, 1]$, define $T(x) = \mathbb{E}[y| F(x)]$ to be the ...
- 1
0
votes
0
answers
18
views
Expected Value / Variance of Gamma to the Negative Integer
For random variable $Z$ from a $Gamma(p), p > 0$ distribution we know that the expected value of $E[Z^s]$ is simply the gamma function at $p+s$ divided by the gamma function at $p$, for $p+s$ > ...
- 33
1
vote
1
answer
68
views
What is the fourth moment of a Euclidean Norm?
Let $X=\lVert M^\top p\rVert_2$, where $M$ is an $n\times n$ non-random matrix and $p\sim N(0,I_{n\times n})$ is an $n\times 1$ vector, and$\lVert \cdot\rVert_2$ is the Euclidean norm.
Using some ...
- 1,166
2
votes
3
answers
74
views
Baffled by Rubin's Potential Outcomes RE: What Would Have Happened?
Background
Seeking a clear and authoritative explanation of a key concept of Rubin's Potential Outcomes Framework that is causing this hapless OP enormous grief. While the necessity to distinguish ...
- 121
0
votes
1
answer
72
views
How to tell if an estimator is unbiased? How to find expected value of an estimator?
You come up with a great idea of an estimator for $\beta_1$ in the SLR model which satisfies SLR.1 to SLR.4:$$y_i=\beta_0+\beta_1x_i+u_i$$ Given a sample $\left\{(x_i,y_i),i=1,2,3,\dots,n\right\}$, ...
3
votes
0
answers
87
views
Expected value for a fair die game
You roll a fair die until the cumulative sum rolled so far is a multiple of 3. You get $1 for each roll. What is the expected amount you will get?
I started by finding the probability of the game ...
- 171
2
votes
1
answer
210
views
Monte-Carlo integration with importance-sampling
I came across a paper, where (section 3.2) importance sampling is used to estimate an integral. I think I understand what importance sampling is but I don't understand how they got the solution.
The ...
- 23
1
vote
2
answers
196
views
Dice rolling problem
A game is played with $n$ dice, and an additional parameter $r<=6$. The game has one player who must throw all $n$ dice on each round and add the score to the total. The game works as follows:
The ...
- 51
0
votes
0
answers
38
views
Given two rvs $X$ and $Y$, if $X Y = Z$, is it possible to change the mean and sd of $X$ without changing the mean and sd of $Y$ and $Z$
I have two lognormal rvs $X$ and $Y$, and a third rv $Z$ which is the product of the former two. I know the mean and standard deviation of the three.
Is it possible to calculate an alternative pair of ...
- 113
0
votes
0
answers
10
views
Calculating Best Linear Predictor: vector transposed translated into non-transposed vector?
I am studying the best linear predictor part of Conditional Expectation and the Projection part about the regression model.
while we are looking for the Beta that minimizes S(B), I quite do not ...
- 1
8
votes
2
answers
171
views
Is there anything interesting to be taken from the fact that $E[(X-E[X])(Y-E[X])] = E[(X-E[X])(Y-E[Y])]$?
While playing around with the formula for covariance, I discovered something I wasn't expecting. Replacing the $E[Y]$ in the definition of covariance with an $E[X]$ appears to simplify back down to ...
- 226
0
votes
1
answer
17
views
Expected index of a random element in a list of expected length 1+\alpha
I have a list $l$ with $E(|l|)=1+\alpha$, where $\alpha>0$.
I choose a random element uniformly from $l$ and want to know what is the expectation of the index of this element in the list.
If $\...
- 115
1
vote
1
answer
39
views
Expected number of elements in random bucket, where the probability of a bucket is proportional to its size
I distribute n balls into m buckets randomly and uniformly. Let's assume that n/m is greater than 1, and m is a large number.
Next, I choose a bucket at random, but the probability of selecting a ...
- 115
1
vote
2
answers
60
views
Product of two random variables (one continuous, one discrete)
I am interested in computing the following expected value: $E[yz]$, with $z$ taking values 0 or 1 so that $E[z]=Pr(Z=1)\equiv p$. The random variable $y$ is continuous and in general correlated with $...
- 693
0
votes
0
answers
27
views
Relationship between Ratio of expectation squared vs ratio of squared expection
I have these pair of numbers
$ (a, b) = (\frac{4}{9}, \frac{1}{9}) $ and $(c, d) = (\frac{1}{2}, \frac{1}{6}) $.
Note that - (a, b) are pair of numbers which represent $((E(e_1))^2, (E(e_2))^2) $ and (...
- 143
2
votes
2
answers
160
views
How should I interpret the assumption of the regression?
I read an econometrics book which states one of the basic assumptions of regression is that
$$E(u|x) = 0$$
In another book however I see it written that
$$E(u_i|x_i) = 0$$
Are these two saying the ...
2
votes
3
answers
327
views
Expected value and variance of median
Suppose $Y|\Lambda\sim U(0,\lambda)$ with $\Lambda \sim U(0,1)$. If there is sample with size $n$ of $Y$ (To simplify, assume $n$ is odd, so $n=2m-1$). How do I calculate the expected value of median (...
- 23
2
votes
1
answer
80
views
Is the Law of Large Numbers Related to the Occurrence of Rare Events Over Many Trials?
I recently watched an episode of "The Big Bang Theory" where Sheldon makes a comment about the Law of Large Numbers. In the episode, Sheldon realizes he needs eggs, and almost immediately ...
- 23
0
votes
0
answers
46
views
Variance of powers of a standard normal random variable
To predict growth of money in a stock market I try to calculate expected return over a longer timeframe (e.g. 30 years) with a confidence interval. The simple math of taking an average stock market ...
- 139
0
votes
1
answer
48
views
Convergence of $E(|X|^r)^{\frac{1}{r}}$ [closed]
For a random variable $X$ on $[0, 1]$ with $F(1) = 1$ and $F(x) < 1$ for all $x < 1$, show that $E(|X|^r)^{\frac{1}{r}} \to 1$ as $r → ∞$. If $F$ is such that $F(x) < 1$ for all $x ∈ \mathbb{...
- 281
0
votes
1
answer
33
views
Between steps for fisher information matrix element using Poisson regression?
I am currently working through some math related to my work, and trying to understand how the individual pieces of the following equations come together for the Fisher information matrix expression (...
- 3
0
votes
0
answers
46
views
Variation on St. Petersburg Paradox, with total loss at the end
I’m not too sure how to answer these variations I thought of and was hoping someone could enlighten me.
A casino offers a game of chance for a single player in which a fair coin is tossed at each ...
- 1
3
votes
1
answer
87
views
Let $N(t)$ is a poisson process with rate $\lambda$, $T^* \sim \operatorname{Exp}(\lambda^*)$, find the expectation of $N(\min(t, T^*))$
Currently, my approach is to split $N(\min(t, T^*))$ like the following by the law of total expectation.
\begin{align*}
&E(N(\min(t,T^*))) = E(N(t \wedge T^*)) \\
= {}&E(N(t\wedge T^*) \...
- 133
7
votes
1
answer
198
views
Expected absolute deviation greater than standard Laplace
Could there exist a distribution, other than standard Laplace (probability density of the form $1/2e^{-|x|}$), on $\mathbb{R}$ such that $E[x]=0,E[|x|]=1$ and that
\begin{equation*}
E[|x-a|] \geq |a|+...
- 145
3
votes
1
answer
53
views
Reducing Variance in Estimating the Exponential Average of Random Variables
Imagine we have a random variable called X, and the function form of the probability density for X is unknown. Now, I'm interested in finding the average value of the exponential of X, denoted as E[...
- 131
3
votes
1
answer
63
views
Tossing Until First Heads Outcome, and Repeating, as a Method for Estimating Probability of Heads
Consider the problem of estimating the heads probability $p$ of a coin
by tossing it until the first heads outcome is observed. Say we get $k_1$
tosses, then $U_1 = \frac{1}{k_1}$ is an estimate for $...
- 33
2
votes
0
answers
51
views
Count distribution in which the mean is equal to the standard deviation?
I have a data set with counts that exhibit the sample property that
$$\hat \sigma_x \approx \bar x$$
which is to say that the sample standard deviation (with Bessel's correction) appears to ...
- 7,118
0
votes
1
answer
39
views
(Using conditional expectation to calculate) expected value of the product of two dependent random variables
Let $\mathbf{X}$ be Binomial point process in $W = [0, 6] \times [0, 4]$ with $n$ points. Let $A_1 = [0, 2] \times [0, 4]$, $A_2 = [0, 6] \times [0, 2]$, and $A_3 = [2, 6] × [2, 4]$. I want to find $E[...
- 1,446