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.

learn more… | top users | synonyms

1
vote
0answers
43 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 ...
5
votes
1answer
50 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: $$ ...
2
votes
1answer
63 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 ...
0
votes
0answers
28 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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
4
votes
2answers
48 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 ...
1
vote
1answer
34 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 ...
-1
votes
0answers
22 views

Expectation of product of random variables whose covariance is unknown

Is it possible to compute $\mathbb{E}[XY]$ where $X \sim \mathcal{N}(\mu_X, \sigma_X^2)$ and $Y \sim \mathcal{N}(\mu_Y, \sigma_Y^2)$ if $\mbox{cov}(X,Y) = \sigma_{XY}$ is unknown? My intuition tells ...
3
votes
0answers
19 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, ...
2
votes
0answers
28 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.
1
vote
0answers
38 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 ...
3
votes
1answer
52 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 ...
0
votes
0answers
22 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 ...
0
votes
2answers
127 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 ...
2
votes
0answers
42 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 ...
1
vote
1answer
88 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 ...
0
votes
1answer
38 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 ...
0
votes
0answers
17 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 ...
2
votes
1answer
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 ...
1
vote
1answer
56 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 ...
1
vote
1answer
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 ...
4
votes
2answers
65 views

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 ...
1
vote
0answers
22 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. ...
0
votes
0answers
17 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 ...
1
vote
0answers
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 ...
3
votes
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 ...
1
vote
1answer
42 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.
0
votes
0answers
17 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
votes
1answer
79 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
votes
1answer
29 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 ...
2
votes
2answers
42 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 ...
3
votes
1answer
32 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: $$ ...
1
vote
1answer
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 ...
3
votes
1answer
43 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 ...
2
votes
0answers
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 ...
1
vote
1answer
59 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 ...
1
vote
1answer
29 views

Expected value of the inverse of a random variable

Let $X$ be a random variable. $X$ can take the value 1 with probability $p$, and the value 2 with probability $1-p$. Can we write $E[\frac{1}{X}] = \frac{1}{E[X]}$? (note that $E[X] \neq 0$) Thank ...
0
votes
1answer
41 views

Study design - fresh look!

Need to be advised outside of the circle. I am more a physiologist+mathematician plus-c,c++,java coder/developer. Chart data. From year 2001 till 2012. 89 nursing stations or emergency call ...
2
votes
1answer
88 views

Expected survival time from log-logistic survival model in R from survreg

I'm currently estimating a survival model (accelerated failure time model) with a log-logistic distribution in R using the survival package and the survreg function. I want to simulate expected ...
1
vote
1answer
45 views

What is the expected number of coin flips, if you stop when the first coin flip is the same as the last?

In order to calculate the $\text{E}[X]$ where $X$ is the number of total coin flips, this is the approach I took: The probabilities are: $Pr(H) = p$ $Pr(T) = (1-p)$ Define indicator random ...
9
votes
1answer
114 views

Show that if $X \sim Bin(n, p)$, then $E|X - np| \le \sqrt{npq}.$

Currently stuck on this, I know I should probably use the mean deviation of the binomial distribution but I can't figure it out.
6
votes
1answer
62 views

Mean of predictive distribution

I observe independent, Poisson-distributed data $ D = \{x_1, ... x_n \} $ with mean parameter $ \mu $, i.e., $$x_i\stackrel{\text{iid}}{\sim}\mathcal{P}(\mu)$$ Over $ \mu $ I assume $ Gamma(\alpha_0, ...
1
vote
0answers
26 views

Find expectation or lower bound of log erf

I need to find the expectation of $\log \Phi(x)=\log \left(\int_{-\infty}^x\frac{1}{2\pi}\exp(-\frac{1}{2}s^2)ds\right)$. (I realise this isn't quite the error function, but not sure what to call it). ...
2
votes
1answer
52 views

Law of iterated expectations with two random variables

Let $X$ and $Y$ be two random variables. I want to calculate $E[X|X<Y]$. I am wondering whether I can use the law of iterated expectations in order to calculate it, i.e. $E[E[X|X<Y,Y]]$. Do I ...
1
vote
0answers
19 views

If $E[X(t)X(s)]=t \land s $.Show that this process has independent increments

Let $X(t), t\ge0$ be a real-valued Gaussian process with mean zero and covariance function $E[X(t)X(s)]=t \land s $.Show that this process has independent increments.
1
vote
1answer
28 views

Why is $E(u^2)=Var(y)$? (Binary Response Model)

I'm trying to show some results in binary response models, and a couple of the proofs use the "fact" that $E(u^2)=Var(y)$, but I can't see why this is. The setup is that $y$ takes on the value $0$ or ...
0
votes
2answers
60 views

Can’t Find the Fisher Information of This Function

Can anyone help me find the Fisher information for this function: $$f(x|\lambda) = \lambda\,x^{\lambda-1}\quad \text{ where } \lambda \in [0,1]\,.$$ Thanks in advance!
3
votes
0answers
35 views

Bounding the expectation of the difference between empirical vs generalization error

Let the (defect) difference between empirical and generalization error be: $$D[f_S] = I_S[f_S] - I[f_S]$$ where the empirical risk is: $$I_S[f_S] = \frac{1}{n}\sum^n_{i=1} V(f_S,z_i)$$ where ...
0
votes
0answers
49 views

Calculate expected value from discrete beta distribution

I have a lookup table in 2 variables, $Z_l$ and $T_l$. So, $Z_l$ and $T_l$ are vectors with same length where $Z_l$ goes from 0 to 1 and $T_l$ varies between 300 and 2000. If you are curious, $Z_l$ ...