Scoring systems aside from expected value for ad conversions?

I am trying to develop a scoring system for ad conversions. I have the type of data you would expect such as click-through rate (proportion) and the dollar amount of the sale. So I know I can develop an expected value scoring system from this type of data. But I want to find is there … Read more

Is the covariance between the product of two variables and one of the variables zero?

For two centered (zero expectation) random variables X and Z I am interested in the covariance of the product XZ and either X or Z. Cov(X,XZ)=E(X(XZ−E(XZ)))=E(X2Z) I think the last quantity can be non-zero. However, for any choice of X and Z I have tried, in simulations, I find the quantity to be approx. zero, … Read more

Team Expected Points Given Very Limited Information

Suppose we were interested in projecting Team A’s expected points in a given game. Lets say, on average over a very large sample size, they score 3 points a game against an average team. Then lets say that Team B allows 2 points a game against an average team. Finally, we will say that the … Read more

Zero element in the inner product space of all random variables with finite variance and $\langle X,Y\rangle = E(XY)$

I want to prove that the set of all random variables with finite variance and $\langle X,Y\rangle = E(XY)$ is an inner product on this set. I want to see that $\langle X,X \rangle = 0$ iff $X = 0$. My first problem is that I’ve never been told what the zero element of this … Read more

𝔼[exp(−α‖X‖)] when X is gaussian vector

Suppose X is a d-dimensional random vector where $X \sim N(\mu, \sigma^2 I_d)$. What is $\mathbb{E}[\exp(-\alpha \|X\|)]$ ,where $\alpha>0$? Since norm is not squared, it can’t be factorized and we can’t take advantage of independence. Otherwise, it was easy to solve. Any help is appreciated. Answer AttributionSource : Link , Question Author : mathlover , … Read more

Expected value of neural network output

I’m attempting to find a closed form expected value of a particularly messy expression involving dot products of multiple neural network outputs. A sub-problem of this is to consider just a single feed-forward network. Consider the feed-forward neural network with weight matrices $W_1 \in \mathbb{R^{k_2 \times k_1}}$, $W_2\in \mathbb{R^{k_3 \times k_2}}$, and input $x\in \mathbb{R^{k_1}}$. … Read more

Tricky Conditional Expectation

A group of 21 women and 14 men have a certain disease with probability p, independently. If we know that exactly 5 of the people have the disease, what is the expected number of women who have the disease? Using LOTE here, we can assign an indicator random variable for those with the disease and … Read more

Derive Expectation/Variance of Histogram

Derive the expectation and variance of the following: $\frac{1}{n(b-a)}$$\sum_{i=1}^n 1$$_{Y_i∈(a,b]}$, b>a. My thoughts with this are to put the summation in an integral and then multiply by n. Is this a correct way of thinking? Is a moment generating function needed? Answer AttributionSource : Link , Question Author : Mike W. , Answer Author : … Read more

Fourth moment of complex Gaussian r.v

Let x∼CN(0,a) (a complex Gaussian random variable) and I know that the p.d.f. of z is defined by: p(z)=1√2πae−|z|22a. Given that, how can I calculate E[|z|4]=? Could someone show how to get E[|z|4] step by step with the integrals, please? Answer AttributionSource : Link , Question Author : Felipe Augusto de Figueiredo , Answer Author … Read more

How many samples are needed to approximate the standard deviation?

Assume you have a random variable X for which you want to compute a 1/4-approximate σ′ of σ=√Var(X) (i.e. |σ−σ′|≤σ/4). You can only get samples drawn according to X (you have no other information on X). My question is: how many samples do you need to achieve this task (with constant success probability)? If the … Read more