# Tagged Questions

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### Infinite fourth moment and maximum entropy

Alright, I expect this is a silly question, but I don't actually know, so. Suppose there is some random variable that's distributed on the reals, and all I know about the distribution is its mean ...
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### Proving independence of variables with normal distribution [on hold]

Random variable $X$ is a variable with standard normal distribution. How to prove that $|X|$ and $\frac{X}{|X|}$ are independent? Thanks.
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### Convergence in distribution for changing domains.

I am trying to consider whether this is possible and/or reasonable: Let $X_n:\Delta_n \to \mathbb{R}$ be a sequence of random variables, defined over a unique space $\Delta_n \subseteq \Omega$ for ...
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### Probability density function of two uniformly distributed stochastic variables

I'm currently stuck on an exercise involving two independent stochastic variables X and Y. Both X and Y ~ U(0,1) (uniform distribution) The goal of the exercise is to calculate the probability ...
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### Question about a change of variable used to compute $E(X)$ from the CDF of $X$

I was studying a proof where the author shows that if the range of x is $\mathbb R_+$ and $F$ is the cumulative distribution function then: $$E[X] = \int_{0}^\infty (1-F(x))dx$$ However on one ...
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### probability of a flipped coin

A fair coin is flipped three times. Let $A$ be the event that a head occurs in the first flip and $B$ be the event that exactly one head occurs. a) Find $p(A/B)$ b) Are $A$ and $B$ independent? ...
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### If pages in a book have an iid Poisson number of errors, in 10 pages what is the probability that exactly 3 pages have exactly 1 error?

Suppose the number of spelling error on any given page in particular book can be modeled by a Poisson distribution with $\lambda=2$, and assume that the number of errors on different pages is ...
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### Choosing random marbles until one is divisible by $X$ [closed]

A box contains twelve marbles on which they are numbered by $1,2,3,...,12$. Now let $X$ represent the number of marbles you must choose with replacement until you obtain one with a number that is ...
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### Sum of normally distributed independent random variables, where one has a different (exponential) unit

$$X \sim \mathcal{N}(\mu_X,\,\sigma_X^2)$$ $$Y \sim \mathcal{N}(\mu_Y,\,\sigma_Y^2)$$ $\mu_X$ and $\sigma_X$ have unit decibel watt ($\text{dBW}$); $\mu_Y$ and $\sigma_Y$ have unit watt ($\text{W}$). ...
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### Polynomial joint pdf $f(x,y)$ such that of $f(x) \neq f(y)$

How can I build a polynomial joint pdf $f(x,y)$ for $x \in [x_1, x_2]$ and $y \in [y_1, y_2]$ such that of $f(x) \neq f(y)$ or equivalently, $x$ and $y$ are depended on each other?
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### What is the difference between $E[X\mid Y]$ vs $E[X\mid Y=y]$ and some of the properties of $E[X \mid Y]$?

I was trying to understand both intuitively and rigorously what the difference between $E[X\mid Y]$ vs $E[X\mid Y=y]$. Let me tell you first the things that do make sense to me. $E[X\mid Y=y]$ makes ...
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### How to estimate the covariance matrix if the unnormalized pdf is known but integral is intractable? [duplicate]

Assume a $d$-dimensional random vector $x$, whose unnormalized pdf is known as the product of N multivariate t-distribution: $$Pr(x)\propto\prod_{i=1}^nt_{\nu_i,\mu_i,\Sigma_i}(x)$$ Is there any ...
Can one show (and how) that $$Pr[X \gg Y]\approx 1$$ for $$X:=\sum_{i=1}^k Bin\left(n\left(\frac{1}{2}\right)^i,i\right)$$ and $$Y:=\sum_{i=k+1}^{\infty} ... 0answers 63 views ### Prove Pr[X + Y \geq x] \sim Pr[X \geq x] We have two independent random variables X_n and Y_n, where$$X_n=\sum_{i=0}^n x_i$$and$$Y_n=\sum_{j=0}^n y_j,$$where x_i,y_j are (non-identically) Bernoulli distributed and independent. ... 2answers 76 views ### Partial sum of binomial I 'm trying to figure out a closed form solution for the following summation: \sum_{j=0}^{\omega} j{n \choose j}p^{j}(1-p)^{n-j} where \omega < n Is there any closed form solution? 1answer 118 views ### Averaging inverse CDFs Suppose I have two distributions P and Q on the line that admit well defined inverse cumulative distribution functions F^{-1}_P and F^{-1}_Q. I define an "average" distribution A as the ... 1answer 40 views ### Expected Payment under limited policy The unlimited severity distribution for claim amounts under an auto liability insurance policy is given by the cumulative distribution:$$ F(x) = 1 - 0.8e^{-0.02x}-0.2e^{-0.001x} , x \geq 0$$... 1answer 35 views ### Question about exp. distribution We know that X\sim \exp(1),Y\sim \exp(2) and they are independent. What is P(Y>X)? exp=Exponential... Thank you! 2answers 29 views ### What is Cumulative Binomial probabilities? I am new to this so don't know if I am asking the right question as I just read about its usage but didn't know what exactly a Cumulative Binomial probability is. So my question is, What is ... 0answers 42 views ### Is the following probability distribution stationary/constant For a conservative system, we know that angular momentum, l, and total energy, E, are constant, i.e. \dot{l}=\frac{dl}{dt} = 0 and \dot{E}=\frac{dE}{dt} = 0, where t indicates time. Let ... 1answer 39 views ### Find Limiting Distribution of |X_n| Let Z_1,Z_2,...,Z_n,... be a sequence of independent standard normal random variables. Let X_n=\sum^n_{k=1}\frac{Z_k}{\sqrt{k}}. Does the limiting distribution of |X_n| exists? If yes, find it; ... 2answers 31 views ### We are making a Bernoulli experiment… [on hold] We are making series of independent Bernoulli experiment with \frac13 chance to success. What is the probability that we got success at the first experiment, if we know that we get two successes at ... 1answer 34 views ### Method of moments for Beta (\alpha_1,\alpha_2) distribution I am trying to solve for the first two moments of a Beta(\alpha_1,\alpha_2) distribution. We know that the first moment is equal to: \mu_1 = \frac{\alpha_1}{\alpha_1+\alpha_2} and the second ... 1answer 56 views ### Prove Number of Arrivals N(s) up to time s follows \mathrm{Poisson}(\lambda s) Distribution This comes from my self-study of Durrett's "Essentials of Stochastic Processes" book, page 97. Definition Let \tau_1,\tau_2,\ldots be independent \mathrm{exponential}(\lambda) random variables. ... 1answer 24 views ### Asymoptotic distribution of identically distributed random variables [closed] Y_1, Y_2, ..., Y_N are independent and identically distributed random variables with the distribution function F := F_{Y_1} and F'_n(y) = \frac{1}{n}\sum_{i=1}^{n}\mathbf{1}_{\{Y_i \leq x\}} as ... 0answers 15 views ### Probability density function of an element How to find the probability density function of x_m\left(1\le m\le n\right) from joint density function, p_X\left(x_1,x_2,\cdots,x_n\right), of n random variables which satisfy following ... 0answers 12 views ### Error of a Serial Processs Give random variable X and two processes A, B . Assume that  Y_{1}, Y_{2} are estimated versions of X by using processes A, B respectively, with probability: P\left \{ \left | X-Y_{1} \right ... 2answers 86 views ### Easy way to compute Pr[\sum_{i=1}^t X_i \geq z] We have a set of t independent random variables X_i \sim \mathrm{Bin}(n_i, p_i). We know that$$\mathrm{Pr}[X_i \geq z] = \sum_{j=z}^{\infty} { n_i \choose j } p_i^j (1-p_i)^{n_i -j}.$$But is ... 0answers 16 views ### generate probability distribution from histogram let us consider following code ... 1answer 29 views ### derivation law from the call option formula i am reading a article about the option pricing. and i got stuck with some typical statement. C(K)=\int (x-K)^+\mu(dx) is given. here, \mu is implied law of asset price and C(K) is the price ... 0answers 22 views ### X and Y are i.i.d random variables with finite second moments. X+Y and X-Y are independent, show that X is Gaussian. X and Y are i.i.d random variables with finite second moments. X+Y and X-Y are independent, show that X is Gaussian. Without loss of generality we may assume that X and Y are ... 1answer 71 views ### [Probability]need help to understand the following expression So assume Y and X are exponentially distributed with parameters y_1, and x_1 respecitively. assume c is a constant. I am having huge trouble to understand the integration of the following ... 0answers 19 views ### Kullback-Leibler or Jensen-Shannon divergence between two distributions. i would like to understand well what Kullback-Leibler or Jensen-Shannon divergence between two distributions will tels us about two distribution,for instance let us consider following code ... 0answers 15 views ### Are functions of independent random variables related to each other by a constant independent I have 6 random variables a,b,c,d,f,g, each having independent Gaussian distribution. Now I define following three random variables X = ab - cd\\ Y = cf - ag\\ Z = gd - bf ... 1answer 35 views ### Measurability and knowledge there seems to be a subtle relationship between knowledge and measurability. If I have a stochastic process (X_n)_n, then for example a stopping time ( other examples would be martingales, ... 0answers 30 views ### Probability: NEED HELP to Understand with the follow [duplicate] I need help to understand the probability derviation of a paper. Please help me. For the following, please only treat |h_{R,B}|^2 and |h_{A,R}|^2 as random variables (other parameters can be ... 1answer 23 views ### Convergence in total variation There are the very basic convergence types in probability theory: almost sure, in L^p-norm, in measure and in distribution. Besides that there is the concept of convergence in total variation norm. ... 1answer 26 views ### Product of 2 random variables:domain of integration I am trying to compute the PDF of the product of two ind. random variables: Z=XY, where 0\leq x \leq d and  0\leq y \leq 1 . (0<d<1) I found this formula :  f_Z(z) = ... 1answer 64 views ### Exponential Distribution question I'm having some trouble understanding the mechanics of how to solve with this distribution. The question: The number of years that a washing machine functions is a random variable whose hazard rate ... 0answers 35 views ### What is the variance of an arbitrary “good” function of several independent normally distributed random variables During my studies years ago I came over a formula that states something like if x_i are independent normally distributed variables with variances \sigma^2_i and f(x_i) is differentiable (and ... 1answer 36 views ### How to prove that convergence in MGF implies Convergence in Distribution? I know that if the moment generating function of two distribution converges to the same function then the two distribution converges in CDF. But how can we prove this thing explicitly ? 1answer 50 views ### Extreme Value Theory - Show: Normal to Gumbel The Maximum of X_1,\dots,X_n. \sim i.i.d. Standardnormals converges to the Standard Gumbel Distribution according to Extreme Value Theory. How can we show that? We have$$P(\max X_i \leq x) = ...
I have four random variables $X_1$, $X_2$, $X_3$ and $X_4$. Their joint dist. is $f(x_1,x_2,x_3,x_4)= \exp(-x_1-x_3)$, where limits are $x_4 = 0$ to $\infty$, $x_3 = x_4$ to $\infty$, $x_2 = x_3-x_4$ ...