# Tagged Questions

Questions about maps from a probability space to a measure space which are measurable.

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### Variance of sum of linear combination

I want to calculate the variance of a sum of linear combinations, so $$\operatorname{Var}\left(w'R_1 + w'R_2\right)$$ where $w$ is a $N\times 1$ vector and both $R_1$ and $R_2$ are $N\times 1$ ...
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### $\mathbf{E}\left[\frac{(U_1+c)^2}{\max((U_1+c)^2, U_2^2)} \right] \ge \mathbf{E}\left[\frac{U_2^2}{\max((U_1+c)^2, U_2^2)} \right]$

We consider two i.i.d. random variables $U_1$ and $U_2$ such that $\mathbf{E}[U_1] = \mathbf{E}[U_2] = 0$ and $\textrm{Var}[U_1] = \textrm{Var}[U_2] < \infty$. Prove that for any $c > 0$ the ...
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### Which probability is greater, given minimal info

which probability is greater, given that $X$ and $Y$ are independant, positive random variables? There is also the option that it's impossible to know as we don't have enough information. I'd ...
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### $\lim X_n = 0$ iff $b > 0$

Probability with Martingales: It looks like $$\lim \exp\{aS_n - bn\} = 0$$ if $b > 0$ because $$\lim aS_n - bn = -\infty \tag{*}$$ but how to prove $(*)$?
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### Independence of linear combinations of random variables

This seems like a straightforward question, but I'm having trouble finding anything on it. Suppose we have a set of random variables, $X = (X_1,...,X_p)$ (the components of which may not be ...
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### Maximize sum with no two consecutive variables

Random variables $x_1,x_2,\dots,x_{100}$ are drawn independently from the uniform distribution over $(0,1)$. After knowing the values, we are allowed to choose a subset of them as long as no two ...
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### How to create a variable which changes randomly and smoothly?

I want to create a variable which is assumed to be the acceleration of a car. I assume it should has zero mean and normal distribution. But the acceleration cannot change rapidly. How do I make it ...
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### Variance of the sum of correlated variables

If the variance of two correlated variables is: $$Var(r_1+r_2)=\sigma^2_1+\sigma^2_2+2\textrm{cov}(r_1,r_2)=\sigma^2_1+\sigma^2_2+2\rho\sigma_1\sigma_2$$ where $r_1$ and $r_2$ are vectors, then what ...
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### Shifting the mean of a composite function of deterministic and random variables

For a project I am involved in relating to communication, I have the following model: $L = f(r).X$ where $X$ is a lognormal random variable with zero mean in the logarithmic scale and standard ...
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### Transformation of random variables that preserves the distribution

Suppose we have a random variable $X$ with distribution $F_X$. Let $X_1$ and $X_2$ be two independent copies of $X$. My question: can we find a transformation $Z=g(X_1,X_2)$ such that the ...
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### Expected value and variance of a random variable, defined as the largest of $6$ randomly drawn numbers

Let each of the numbers from $1$ up to $49$ be written on a ball, and let all these balls be contained in a box. From this box, we randomly draw exactly $6$ numbers (without putting them back, so we ...
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### Relationship Between $\mathbb{E}$(time) and $\mathbb{E}$(Repetition)

Consider aa Stochastic Process with Expected value of time of occurring =T (less than infinity). Can we deduce that Expected value of number of occurrences until time T is equal to 1?? If not, in ...