# 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.

1,672 questions
Filter by
Sorted by
Tagged with
0answers
22 views

### Graphical representation of unconditional expected value

First, let tell you that I've being struggling with the concept of unconditional expectation for linear regression. For conditional expectation is easier: We know that the conditional expectation ...
1answer
48 views

### Expected value is known

I'm a agronomy student in Colombia and I've been recently studying from the book Generalized Linear Models With Examples in R by Dunn and Smyth. As you can imagine, I do not have a pretty good ...
5answers
2k views

### Probability that number of heads exceeds sum of die rolls

Let $X$ denote the sum of dots we see in $100$ die rolls, and let $Y$ denote the number of heads in $600$ coin flips. How can I compute $P(X > Y)?$ Intuitively, I don't think there's a nice way to ...
1answer
112 views

### The expected value and variance of E(-1X)? [closed]

This might be a stupid question, but how I can calculate the expected value $\operatorname{E}(-1X)$ and variance $\operatorname{Var}(-1X)$ for example in a case in which $X\sim N(100,0.1^2)$?
1answer
114 views

### Likelihood of my friend being able to guess skittle taste

I'm preparing for a data science interview, and here's a question I encountered during my preparation: Your friend claims he can tell the five colors of skittles apart by taste alone. The probability ...
1answer
33 views

### Understanding covariance

I came across following problem: A discrete random variable $P$ takes values $-3,-2,0,2,3$ with probability $0.2$. Let $Q=P^2$ be another random variable. What is covariance of $P$ and $Q$? I solved ...
2answers
36 views

### Covariance for Random Variables vs. Sample Data

In my textbook, it says that the formula for finding covariance between two random variables is: $Cov(X,Y)=E((X-EX)(Y-EY))$ With $EY$ and $EX$ being the mathematical expectation for the random ...
0answers
37 views

### Expectation of a sequence of random variables based on a set of iid Gaussian random variables

This is a rather convoluted problem: I'll my best trying to explain it. So, we have $m$ iid standard Gaussian RVs $Q_i$. We get a realization from each of them, and these values $q_1,\dots,q_m$ are ...
0answers
11 views

### Help me decipher the notation in the given definition

I was going through a book on statistics and I came across this definition of expectation. Let $X : \Omega \rightarrow D$, be a discrete random variable and let $\phi : D \rightarrow {\rm I\!R}$ The ...
0answers
7 views

### Expectation of truncated autoregressive process

Consider the following nonlinear autoregressive model: \begin{align*} \epsilon_t & = \epsilon^\rho_{t-1} e^{u_t}, \\ x_t & = \frac{\gamma \, \epsilon_t x^a_{t-1}}{1+b y_t}, \\ y_t & = \...
1answer
70 views

### How to derive the expectation of $\ln \mu_j$ in Dirichlet distribution

I have derived the mean and variance of $\mu_j$ in Dirichlet distribution $\text{Dir}(\mu_1, \cdots, \mu_K|\alpha_1, \cdots, \alpha_k)$. On https://en.wikipedia.org/wiki/Dirichlet_distribution, it ...
0answers
73 views

### Upperbound for Norm of Lagged Autocovariance Matrix

Suppose $X_t$ is a vector valued time series in $\mathbb{R}^d$. Assume for the moment that $X_t$ is stationary with $EX_t=0$, $E\|X_t\|^2<\infty$, and let $$C_h = E[X_0 X_h^\top]$$ denote the ...
1answer
21 views

### Estimating Population Total of a Lognormal distribution

Say weāre trying to model spending behavior and it has a lognormal distribution, lognormal(6.4, 0.8) with N=1000 independent observations, a vector named A. Whatās the expected value of the total ...
0answers
6 views

### Relationship between two “loss-like” variables

Suppose I have a discrete set $A \subset \lbrace \theta\in\mathbb{R}^3 : \lvert\lvert\theta\rvert\rvert_2 = 1 \rbrace$ of unit vectors, and let $F: A\times \mathbb{R}^n \to R^k$ such that $F(\cdot ,C)$...
1answer
66 views

### How do you calculate the expected value of $E\left\{e^{-|X|}\right\}$ e.g. for Gaussian X?

If $X$ is a random variable, I would like to be able to calculate something like $$E\left\{e^{-|X|}\right\}$$ How can I do this, e.g., for a normally distributed $X$?
1answer
220 views

### Difference between the expectation of x bar squared and the expectation of x squared

I am trying to understand the derivation of the expectation of the maximum likelihood (MLE) of variance, however I am confused as to what the difference is between $\bar{x}$ and $x$. Below you find ...
1answer
94 views

### expected value of a fishing strategy

Suppose there is a pond with infinite number of fish. Weights of the fish are iid uniform $(0,1)$. We catch fish from this pond with the following rules: Each day we catch at most one fish from the ...
1answer
127 views

1answer
38 views

### Upper Bound and Lower Bound on Means when Distributions are bounded?

Suppose we have two different probability distributions $p, q$ defined on input $x \in [0,1]$. We know that for any value of $x$ in the domain, we have $\exp^{-a} \leq \frac{p(x)}{q(x)} \leq \exp^{a}$...
1answer
58 views

1answer
57 views

### Mean and Variance of dot product of 2 random vectors?

x and y are two vectors of dimension k. Assume that the components of x and y are independent random variables with mean 0 and variance 1. What would be the mean and variance of their dot product, x Ā· ...
0answers
38 views

1answer
58 views

### Counterexample where E(u|x)=0 in a regression model cannot hold in the population?

Edit: Background information: I have two variables of interest, $y$ and $x$ that are linearly related via the following: $y = a + bx + u$, where "$a$" and "$b$" are fixed ...
1answer
43 views

### Find $E[N^2 | N > 2]$ for a frequency distribution

N has probability mass function: $p_o = p_1 =0$ and $p_k = \frac{1}{(e^1-2)k!}$ for $k=2,3,4,...$ I Solved for the pgf of N and got $G(t) = \frac{e^t}{e^1-2}$ How do I calculate $E[N^2 | N>2]$?
1answer
35 views

### What is the following expectation?

What is $E[Y_1I(x<Y_2,Y_2>Y_1)]$, where $Y_1$ and $Y_2$ are non-negative continuous random variables and x is a constant? $I(.)$ is the indicator function. Can this be written as follows? \begin{...
2answers
37 views

### Doubt in derivation of expectation of sample variance

I am studying statistics on my own. Please help me in understanding following Here in evaluation of expectation $E[\frac{(n-1)S^2}{\sigma^2}]$, why $\sigma^2$(population variance) is treated as ...
2answers
37 views

### Exogeneity assumption applied to functions of the design matrix

The context of this question is ordinary least squares. $X$ denotes the design matrix. I would like a proof of the claim ā or a corrected version thereof ā made in this other question that the ...
0answers
27 views

### How do derive the expected prediction error for OLS?

At the end of section 3.2.2 of Elements of Statistical Learning, it shows the following: I am having a hard time deriving this. This is what I have so far: \begin{align} E[(Y_0 - \hat{f}(x_0))^2] ...
1answer
33 views

### Intuition for expectation of discrete random variable that takes positive integers

If $X$ is a discrete random variable that takes values on the positive integers, it is true that $$E(X) = \sum_{k=1}^{\infty} P(X \ge k)\;.$$ I know how to prove this (by expressing the summand as a ...
1answer
32 views

### Doubt in independence of 2 random variables

If 2 random variables are independent, then $f(x,y) = f(x)f(y)$. Is converse true? $F(x,y)=F(x)F(y)$. Is converse true? $E(x,y)=E(x)E(y)$. Is converse true? where $F$ is cdf and $f$ is pdf I recently ...
1answer
40 views

### Expectation of a absolute of centered RV is bounded by root of variance?

I've seen a claim that for a RV x where $E(X)=0$, it holds that $E(|X|)\le\sqrt{Var(X)}$. How can you prove this? (and maybe what is the intuition for this?) Thanks
0answers
48 views

### Show that if $Y$ is another random variable such that $E[X] = E[Y]$ and $V(X) = V(Y)$ then $P(Y \ge a) \le p$

Let $p \in (0,1)$ and $X$ be a random variable such that $P(X=a) = p, P(X=-b) = 1-p$ Show that if $Y$ is another random variable such that $E[X] = E[Y]$ and $V(X) = V(Y)$ then $P(Y \ge a) \le p$ and ...
0answers
9 views

### How many trials are needed to have 99% of all possible successes between n independent elements which can have at most 2 successes each?

I want to calculate the expected time of growth of a group of plants in a videogame. If I have $n$ independent plants and each plant has a probability $p$ of being picked each unit of time $t$ (or ...
0answers
22 views

### Calculating the expected value of the log multinomial probit

I am stuck trying to solve the calculation of an expectation related to the multinomial probit likelihood. Say I have a random vector $\mathbf{F}$ whose components, $F_1$, $F_2$, $F_3$, are ...
2answers
107 views

0answers
24 views

### Creating an expected dataset to compare with observed data to test for genetic interactions

I am trying to define a test to tell if two mutations combined give a greater phenotype than what you would expect by simply adding both. Here is an example (using R): ...

1 2
3
4 5
ā¦
34