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.

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Degeneracy paradox

Say I have a highly biased coin that lands heads with $p_h=0.01$ and tails with $p_t=0.99$, and I flip it $98$ times. The probability of zero heads is ${p_t}^{98} \approx 0.373$. The probability of ...
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29 views

Expectation with transformed random variable

The expected value of a function $f$ of a random variable $x\in R^n$ is defined as $E_x[f(x)]=\int p(x)f(x)dx$ where $p()$ is the pdf of $x$. Assume a function $h: R^m \rightarrow R$. Let $z=Mx$, ...
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23 views

How to calculate probability of bankruptcy for $E>0$ and given $\sigma$?

I will demonstrate the problem with the simple example. Suppose I want to start a casino. The probability theory tells me that the expected value for my games is greater than zero ($E>0$), and ...
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20 views

Help please regarding covariance

I have a question regarding covariance. If we have two independent variables $X$ and $Y$, then is the $Cov(X^n, Y^m) = 0$ with arbitrary values of $n$ and $m$? I know that $Cov(X^n,Y^m) = E(X^nY^m) ...
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17 views

Finding the expectaion with respect to a gaussian measure on “half” of $R^n$

I want to calculate the following expectation: $$ \int_{X_\theta} x\phi(x) dx $$ where $\phi$ is the density of a (not necessarily standard) gaussian distribution and $$ X_\theta = \{ x \in ...
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35 views

Expected value or median of a function defined on sphere with uniform distribution

Suppose $S$ is the sphere of radius one. And suppose $f:S\to \mathbb{R}$ is defined as follows: $$f(x_1,...,x_n)=\frac{1}{x_1^2}+\frac{1}{x_2^2}+\cdots+\frac{1}{x_n^2}$$ I am trying to calculate ...
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96 views

$E(X)E(1/X) \leq (a + b)^2 / 4ab$

I've worked on the following problem and have a solution (included below), but I would like to know if there are any other solutions to this problem, especially more elegant solutions that apply well ...
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29 views

Expected value of $e^{\alpha \sqrt{t-s} \ Z}$

How can I find the expected value of this: $e^{\alpha \sqrt{t-s} \ Z}$, where Z is a standard normal random variable. I know the moment generating function should help me with this, but I can't ...
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1answer
33 views

How does this expected value translate into a conditional variance?

I'm working with a simple local level model in a textbook \begin{align} y_t &= \alpha_t + \epsilon_t, \qquad \epsilon_t \sim N(0, \sigma_\epsilon^2) \\ \alpha_{t+1} &= \alpha_t + \eta_t, ...
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46 views

What is the relationship between the mean of a continuous variable and expected value?

What is the relationship between the mean of a continuous variable and expected value? How are they connected?
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15 views

Transformation Theorem & Piecewise Function

I am a total statistics newbie and I hope that you can help me with the following problem: Let $X \sim \mathcal N\left(\mu, \sigma^2\right)$ be a random variable. Define a new random variable $Y$ as: ...
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24 views

Finding expected order statistics from a normal with known parameters [duplicate]

I'm interested in finding the expected value for the kth ordered observation of a normally distributed variable with known standard deviation, mean and n. Could someone let me know the formula for ...
2
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1answer
33 views

Chi-squared distribution and dependence

We know that for a group of independent random variables $\{X_i\}_{i=1}^n$ s.t each $X_i$ is distributed as a chi-squared distribution with degree of freedom one ($\chi_1^2$), the sum of these random ...
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14 views

Joint Gaussian Distribution Question

Drawing a pair of $(x, y)$ from a joint Gaussian distribution with $r$ covariance. Knowing the standard deviations of $x$ and $y$ and knowing $z = x + y$, what is your best guess for $x$?
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73 views

An 'easy' exercise on conditional expectations and filtrations

I am struggling with the following exercise in the context of modeling information structure via filtration to evaluate contingent claims. I hope that someone can explain me how to derive the ...
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52 views

Upper Bound on $E[\frac{1}{1-X}]$ where $E[X]=a$ and $0<a<1$

$X$ is a discrete random variable that can take values from $(0,1)$. Since $\varphi(x)=1/x$ is a convex function, we can use Jensen's inequality to derive a lower bound: $$ ...
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1answer
74 views

Expected value of Y = (1/X) where $X \sim Gamma$

I'm having some confusion over this statement here. Let $T_i \sim Exp(\lambda + \theta)$ and if they are all iid then $\sum_n T_i \sim Gamma(\alpha = n, \beta = 1/(\lambda + \theta))$ I want to find ...
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42 views

Exponential distribution of the time between recurrences

Patients in a hospital are treated for certain illness. The time in days that no recurrence takes place follows exponential distribution with mean of $\theta = 27$ days. The following table gives the ...
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6 views

computing expectations in variational updates

I have a complete log-likelihood expression as follows: $$ L = \sum_{i=1}^N \log P(y_i|x_i, w_i, \beta) + \log P(\beta) + \sum_{i=1}^N \log P(w_i) $$ Now, I need to compute the expectation of these ...
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7 views

difference between sampling stopping times

Let $\{a_n\}$ be an increasing sequence of nonnegative numbers. Let $Z_i$ be a sequence of iid rvs. Define $T_1=\inf\{n\ge1: \sum_{i=1}^n Z_i\ge a_n\}$ and $T_2=\inf\{\text{ODD}\,\,n\ge1: \sum_{i=1}^n ...
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59 views

Expectation maxmisation algorithm increases true likelihood at each iteration

I've heard that the EM algorithm ensures that the true likelihood is non-decreasing at each iteration of the algorithm, but I'm not sure why this is the case. I've provided a basic plot which I ...
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1answer
38 views

Simple question about finding conditional expectation

Let $X,Y \sim U[0,1]$ ($X,Y$ are independent), we want to find $E[X|X>Y].$ I tried a few approaches to the above problem, but am not confident in my answer. One approach is as follows. Note that ...
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23 views

Modelling of probabilistic vs deterministic systems

The learning problem in Statistical Learning Theory is defined as: $$ R(f) = \int_{X,Y} L(y, f(x))P(x,y)\mathrm{d}x\mathrm{d}y $$ where $R(f)$ is the expected risk $L$ is the loss function $P(x, ...
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34 views

Expectation of cube of summation of independent random variables

Where would I begin on this problem? I know I begin with pulling $c^3$. Where would I go from there? And I know that $\mathbb{E}[X] = x_1p_1 + .... x_n p_n$ I'm stuck on the rest, however.
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42 views

Deriving the probability function from a logistic regression model

First off, I don't know a lot of stats. That said, I am hoping you can help me derive the function to calculate the fitted probability from the summary of a logistic regression below: My instinct ...
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1answer
55 views

Continuous random variable and median problem

The definition of median, $m$, for a continuous random variable $X$ is $$P(X\leq m)=P(X\geq m)=\int_{-\infty}^m f(x)\text{d}x=\int_{m}^{\infty}f(x)\text{d}x=\frac{1}{2}$$ where $f(x)$ is the pdf of ...
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25 views

Expectation of output of an LTI system w.r.t. a WSS random process

Let $X(t)$ be a wide-sense stationary random process―i.e., its expectation is a constant and its autocorrelaton function is a function only of time differences―and let $Y(t) = X(t) * h(t)$ where ...
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152 views

Expected number of tosses till first head comes up

Suppose that a fair coin is tossed repeatedly until a head is obtained for the first time. What is the expected number of tosses that will be required? What is the expected number of tails that will ...
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48 views

Is there a formula for a general form of the coupon collector problem?

I stumbled across the coupon collectors problem and was trying to work out a formula for a generalization. If there are N distinct objects and you want to collect at least k copies of each of any m ...
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92 views

Show that $E[\mu_i^2|X_i]=\sigma^2$ and $Var(\mu_i|X_i)=\sigma^2$

I'm a little stuck on this review problem so help would be greatly appreciated! Q: We have the regression model $Y_i=\beta_0+\beta_1X_i+\mu_i$ and we assume that the expected errors are $0$. We also ...
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43 views

How to solve this Expectation of log of random variable

This may seem a trivial Question but I am confused and never come across this kind of expression where I need to simplify for a function of a random variable $R$. I have an expression $E\bigg ...
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19 views

Learning Expected value

We probably have played the game "Throwing Balls into the Basket". It is a simple game. We have to throw a ball into a basket from a certain distance. One day we were playing the game. But it was ...
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19 views

Developing a heuristic for maximizing the “covering” of a distribution

Context There's a board-game called Settlers of Catan in which players compete to be the first to gain 10 victory points by trading various resources in exchange for pieces (or cards) worth victory ...
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1answer
58 views

Algebraic manipulation of $Var(Y|X)=E[(Y-E(Y|X))^2|X]$

Q: Show that $Var(Y|X)=E[(Y-E(Y|X))^2|X]$ is equal to $Var(Y|X)=E[Y^2|X]-(E[Y|X)]^2$. Answer: I know I have to use the law of iterated expectation to get to the second statement but I am having ...
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30 views

Penalty Shootout and Expected Value

There are 3 expert players(A,B and C) in a penalty shootout in a football team. The coach often has difficulty selecting an expert penalty shooter from the three expert players. Therefore, he makes a ...
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If $E[Y|X]=a$ for some constant $a\neq 0$, then does $cov(X,Y)=0$?

I'm currently working on the following problem: Q: If $E[Y|X]=a$ for some constant $a\neq 0$, then does $cov(X,Y)=0$? Now I am quite lost as to how to do this problem as the question does not ...
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23 views

Estimate E[x|A,B]: alternatives to bucketing for non-parametric estimation

I have a set of products. I would like to estimate Expected Value of items sold of the products wrt product price and age of the purchaser. One alternative is to assume a distribution and fit it. ...
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18 views

Finding a consistent estimator mathematically

This is my first post on this website so hopefully everything will go smoothly. Let me first ask the question, then go over my problem. Q: Suppose that we are given $({X_{1i},X_{2i}})$ which is a ...
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24 views

Solving a problem related to expected value

After being all out for 58 and 78 in two matches in the most prestigious tournament in the world, the coach of a certain national cricket team was very upset. He decided to make the batsmen practice a ...
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1answer
21 views

Estimating counts from sampled data

I am working on counting events from sampled web logs. To formalize the problem, consider a random process in which we randomly record an event with known probability $r$. Say we have $n$ recorded ...
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1answer
46 views

KL divergence between a gamma distribution and a lognormal distribution?

Is there a closed-form formula for the KL divergence $D_{KL}(X,Y)$ where $X \sim \mathrm{Gamma}(k,\theta)$ and $Y \sim \mathrm{LogNormal}(\mu,\sigma^2)$ ? Many thanks.
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18 views

Expected value of multiple events

There are three offers Offer A - $ 5 probability of redemption A - P (A) = 0.5 Offer B - $ 4 Probability of redemption B – P(B) = 0.6 Offer C - $ 3 probability of redemption C – P(C) = 0.7 If I ...
3
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1answer
84 views

Find the expectation and covariance of a stochastic process

The problem is: Let $W(t)$, $t ≥ 0$, be a standard Wiener process. Define a new stochastic process $Z(t)$ as $Z(t)=e^{W(t)-(1/2)\cdot t}$, $t≥ 0$. Show that $\mathbb{E}[Z(t)] = 1$ and use this result ...
3
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1answer
30 views

Question regarding covariance

I'm trying to prove a theorem, where it is given that each $X_i$ is independent and identically distributed with mean $\mu$ and variance $\sigma^2$. Within this theorem, I have multiple sub-results to ...
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2answers
43 views

How to calculate $E[X^2]$ for a die roll?

Apparently: $$ E[X^2] = 1^2 \cdot \frac{1}{6} + 2^2 \cdot \frac{1}{6} + 3^2\cdot\frac{1}{6}+4^2\cdot\frac{1}{6}+5^2\cdot\frac{1}{6}+6^2\cdot\frac{1}{6} $$ where $X$ is the result of a die roll. How ...
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1answer
46 views

Deriving K-means algorithm as a limit of Expectation Maximization for Gaussian Mixtures

Christopher Bishop defines the expected value of the complete-data log likelihood function (i.e. assuming that we are given both the observable data X as well as the latent data Z) as follows: $$ ...
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25 views

Expected value of pair of success of

I am working on a problem from Harvard Stat 110 problem set. One of the problem (1.7.a) asks to find $$E\binom{X}{2}$$ , where random variable X is from hypergeometric distribution. What does the ...
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50 views

Expectation of covariance in derivation of Kalman filter

I'm working through the derivation of the Kalman filter equations from this paper (or alternative source here) and I'm unsure of the derivation of the state prediction covariance (equation 2 in the ...
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36 views

AB test - Is it okay to use a result with a low confidence level

Suppose you conduct an A/B test of 10,000 views for each of version A and B, but the results take 3 months to capture. Despite a small number of views, achieving a goal (converting a "view" to a ...
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1answer
64 views

Maximum of uniformly distributed random variables using iterated expectations

I'm working through the problems in Wasserman's 'All of Statistics'. The chapter on expectations and conditional expectations ends with a (seemingly) easy problem: Let $Y$ be the maximum of $n$ iid ...