# Tagged Questions

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

2answers
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### Find the limit of the probability of uniform random variable?

Let $X_1 ,X_2 ,X_3 ,…$ be a sequence of i.i.d. uniform $(0,1)$ random variables. Then, calculate the value of $$\lim_{n\to \infty}P(-\ln(1-X_1)-\ln(1-X_2)-\cdots-\ln(1-X_n)\geq n)?$$ My work: Since ...
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### Convergence of expectations of a sequence of exponential random variables.

Suppose $\{X_n\}$ is a sequence of exponentially distributed random variables, where $X_n$ has mean $1/\lambda_n$. Suppose that $\lim_{n\to\infty}\lambda_n = \lambda>0$. Let $X$ be exponentially ...
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### Differentiating $\int\cdots \int f(X_1,X_2,\ldots,X_n)\varphi_1(x_1,\theta)\cdots\varphi_n(x_n,\theta)~dx_1\cdots dx_n$

Differentiating:$$\int_{-\infty}^\infty \cdots \int_{-\infty}^\infty f(X_1,X_2,\ldots,X_n)\varphi_1(x_1,\theta)\cdots\varphi_n(x_n,\theta)\,dx_1 \cdots dx_n$$ with respect to $\theta$. The result is ...
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### Close-form solution for distribution of the stopping time for a path-dependent random process?

A time series $\{x_s\}_{s=1}^{\infty}$ is generated from $N(\bar{x},1/b)\ i.i.d.$. Parameter $\bar{x}$ is drawn from prior distribution $N(\phi_0,1/a)$. Define conditional expectation of $\bar{x}$ as ...
0answers
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### Sum of Independent Levy RVs is Levy RV [on hold]

I want to show that the summation of independent Levy random variables X and Y with scaling parameters a and b is a Levy random variable with scaling parameter c = (a^(1/2)+b^(1/2))^2 using ...
2answers
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### Does this hold in every case, and if only this one, why? Expectation, mean of random variable.

Characteristic function of random variable $X$ let us denote as $f_X(t)$ and $EX$ it's mean or expectation. Does the following hold in all cases, because it keeps coming up and I don't know why it is ...
1answer
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### Solving inequality of two independent exponentially distributed RVs

I have huge problems solving following excersice: There are two molecules. The decay of the molecules is exponentially distributed with $\alpha_1 = 1$ (for molecule 1) and $\alpha_2 = 2$ (for ...
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### Probability of sum of 10 dice throws [duplicate]

If a die is rolled 10 times. What is the probability that the sum of the results is less than or equal to 20? I was trying to solve this using something like $P(X_1 + X_2 + ....+X_{10} \le 20)$ but ...
1answer
38 views

### Convergence of $V_n=\prod\limits_{i=1}^n U_i$

I struggle to do this exercise: Let $U_1,U_2,\dots$ be a sequence of i.i.d. random variables. We define $$V_n=\prod\limits_{i=1}^n U_i$$ Show that $V_n^{1/n}$ converges almost sure and calculate the ...
1answer
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### Stochastics exam Exercise

The professor uploaded an exam to practice, but unfortunately I have no solutions. Let U be a unifomly distributed random variable on $[0,1]$. 1) Let $X=-ln(U)$. Show that $X$ is distributed ...
1answer
63 views

### The variance of the expected distortion of a linear transformation

Let $A: \mathbb{R}^n \to \mathbb{R}^n$ be a linear transformation. I am interested in the "average distortion" caused by the action of $A$ on vectors. (i.e stretching or contraction of the norm). ...
2answers
36 views

### Transformation of the uniform distribution

I struggle to understand the transformation of a random variable with uniform distribution. For example: Let $X\sim \text{Uniform}(0,1)$ and $T=-2\ln(X)$ and I want to find the CDF of $T$, then I ...
1answer
26 views

### What is the expected distortion of a linear transformation?

Let $A: \mathbb{R}^n \to \mathbb{R}^n$. I am interested in the "average distortion" caused by the action of $A$ on vectors. (i.e stretching or contraction of the norm). Consider for instance the ...
1answer
47 views

### Expectation of $|X-Y|$ when a coin is thrown six times

If a fair coin is thrown six times. Let $X =$ number of heads and $Y = 6-X =$ number of tails. What is $E|X-Y|?$ I was able to come up with this table, but I am not sure if this is correct or not and ...
1answer
38 views

### Distribution of Expectation function into a $|X-Y|$

We know that $E(X+Y) = E(X) + E(Y)$. But why is $E|X-Y|$ $\ne$ $E|X| - E|Y|?$
1answer
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### If Z has a normal distribution with mean 0 and variance $\sigma^{2}$, and $Y=Z^{2}$, what would the density function of Y be? [duplicate]

How would I go about finding this density function? Thanks
0answers
39 views

### Random variable for storing cost to get the target.

There is a simple game for a single player. Player's initial level is $n$ and player want to get level $m$. If player's level became the target level $m$, the game terminates. Player should pay $c_i$...
1answer
22 views

### convergence in probability of division and their expected values

Let $\frac{X_n}{Y_n} \rightarrow 1$ in probability. Then does $\frac{\mathbb{E}[X_n]}{\mathbb{E}[Y_n]} \rightarrow 1$? If not, what are the conditions required for this to hold?
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### Very basic clarifications on sampling processes

Could you help me to clarify some basic notions from sampling theory? Please highlight if anything of what is written below is wrong because I am very confused on the order of logical steps. ...