# Tagged Questions

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### Utility of Probability Generating Function [closed]

The utility of Probability Generating Function , how far known to me , is basically to generate PMF uniquely (what all the popular books of probability have written ) . Now , PGF is constructed with ...
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### How to prove $(X_{n})_{n\in \mathbb N}$ and $(Y_{n})_{n\in \mathbb N}$ are supermartingale and $(Y_{n})_{n\in \mathbb N}$ is convergence to -7

Let $p \in [0 , \frac{1}{2}]$ and $\eta_{i}$ be i.i.d random variables and $P(\eta_{i}=1)=p$ and $P(\eta_{i}=-1)=1-p$ and $\mathcal F_{n}=\sigma(\eta_{1},\cdots,\eta_{n})$ and ...
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### Is Hoeffding's bound tight in any way?

The inequality: $$\Pr(\overline X - \mathrm{E}[\overline X] \geq t) \leq \exp \left( - \frac{2n^2t^2}{\sum_{i=1}^n (b_i - a_i)^2} \right)$$ Is this bound (or any other form of hoeffding) tight in ...
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### Efficient ways to sample from a conditional multinomial distribution

I am currently working on a project where I have to generate samples from a discrete multivariate distribution with uneven support (i.e. one component may take on 2 possible values while another ...
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### Functions of Random Variables

In what cases would I want to know $\operatorname{E}\left[e^X\right]$? I'm in an introductory probability course and I'm presented with exercises like this often, but I'm wondering when someone would ...
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### What is the dimension (or units) of a CDF and PDF?

Given a continuous random variable $X$, what are the units of the PDF and CDF of $X$?
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### Variance of sum of random number of random variables (Cambridge University Worksheet)

In the vein of my last question, I'm now at a roadblock on question 3 of this sheet: http://www.trin.cam.ac.uk/dpk10/IA/exsheet3.pdf (note: it's not my intention to ask every question I get stuck on ...
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### Expectation of Quotient of Sums of IID Random Variables (Cambridge University worksheet)

I'm preparing for an interview which requires a decent knowledge of basic probability (at least to get through the interview itself). I'm working through the sheet below from my student days as ...
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### An inequality for the non-negative random variable

I once read the following inequality Is there any specific name for this inequality? And, how to prove it?
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### If $X_{n+1}$ is a martingale subject to $Y_0,\ldots,Y_n$, then is it a martingale with respect to $Y_0^2,\ldots,Y_n^2$?

I don't have a very solid foundation in measure theory, and this always seems a bit confusing to me so I would appreciate any help. We are given $E \left( X_{n+1} | Y_0,\ldots,Y_n \right) = X_n.$ ...
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### Expected value of a random variable differing from arithmetic mean

I am a student who is taking a random processes class. I have seen that expected value of a discrete random variable is equal to the arithmetic mean of the distribution provided the values it takes. ...
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### Degenerate random variable

Let $X$ and $Y$ be independent $rv$ such that $XY$ is a degenerate $rv$. Can I say that individually $X$ and $Y$ are also degenerate? Why? thanks..
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### Distribution of the hyperbolic distance between random points in the Poincaré disc

Let two points at polar coordinates $(r_i, \theta_i)$ and $(r_j, \theta_j)$ be hyperbolic points in $\mathbb{H}^2_\zeta$ with curvature $K=-\zeta^2$. The radial coordinates of these points are ...
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### Bounding expected maximum of inner products

Let $x=(x_{1},\dots,x_{n})^{T}$ be $n$ dimensional random vector uniformly distributed over the $L1$ unit sphere. That is, all $x$ such that $\|x\|_{1} = 1$ have equal probability and all other $x$ ...
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### The correct probability distribution / way to identify large deviations in a set of daily changes to portfolio value

I am working on a report which is being sent through to end users that should flag to them any "large changes" in the day-to-day values for the past 30 days. These values are day-to-day differences ...
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### Arithmetic relation between expectation values of transformed random variables

Let $X_1$ and $X_2$ be discrete non-negative random variables with $E(X_1) \le E(X_2)$. Assume a function $f$ with the following properties: $f$ is positive, i.e. $f(x) \ge 0$ $f$ is (strictly) ...
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### Product of Sums of Bernoulli variables

Let $x_1 \ldots x_a,y_1 \ldots y_b$ be independent random variables taking values $+1$ or $-1$ with probability 0.5 each. Consider the sum $S = \sum_{i,j} x_i\times y_j$. I wish to upper bound the ...
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### Distribution of a logit transform of a normal variable

If I have a normal distribution with a given mean and variance and apply a logistic transform to it, what is the mean and variance of my transformed variable? This seems like it has to be a well ...
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### Random variables resulting due to single or multiple experiments

Assume there are ‘n’ entities that collect some data after each ‘t’ time instances. Generally their collected data is within an “expected” range. However, occasionally the data exceeds the “expected” ...
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### Probability that a given Normal Distribution is Maximum among others [duplicate]

You are given the mean and standard deviations of N normal distributions x1,x2...xn What is the probability that x1 is maximum? ie. Find P(x1>x2,x3..xn) How do I go about solving this? x1,x2,x3 etc ...
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### How to calculate the distribution of the minimum of multiple exponential variables?

$X_1$, $X_2$, $X_3$ are independent random variables, each with an exponential distribution, but with means of $2.0, 5.0, 10.0$ respectively. Let $Y$= the smallest or minimum value of these three ...
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### Estimating the likelihood of independence of two discrete variables using the co-occurrence count matrix

I have some data about users from different regions visiting different directories of some website. Aggregating that data I get the co-occurrence frequency matrix (for regions and directories). Now I ...
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### How to compute the PDF of a sum of a discrete and a continuous random variable? [closed]

I have a problem with this exercise in probability and statistics: Calculate the probability density function (PDF) of $$Z=X+Y$$ where $Y$ is discrete random variable which is equal to $-1$ and $1$ ...
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### Expected value of modified geometric distribution

I am trying to find the expected value of $X$, where $X$ is the number of orders a customer will make in a lifetime. Assuming that there is a $p=.1$ chance of the customer placing an initial order, ...
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### What is $P(X_1>X_2 , X_1>X_3,… , X_1>X_n)$?

All $X$ are mutually independent and from normal distributions, each with its own mean and variance. If it's easier, $P(X_1 \geq X_i \forall i \in \{1, ..., n\})$ is fine although I suspect it's the ...
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### Product of Independent Gaussian Variables

Let $X$ and $Y$ be two independent normal distributions according to $X\sim\mathcal{N}(0,P)$ and $Y\sim\mathcal{N}(0,Q)$. Is it true to say the following ? ...
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### “Running it” multiple times in No-Limit Hold'em poker

Suppose on the flop in no limit hold'em two players go all-in. They both have 2 hole cards and there's 3 community cards on the board at present. There will be 2 more cards drawn and each player will ...
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### CDF in a discrete simulation (Jerry banks book)

Consider the experiment of tossing a single die. Let $X$ be number of spots on up face of die after toss. Then range space of $X$ is $R_x = \{1,2,3,4,5,6\}$. The discrete probability distribution for ...
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### Composition of probability density

I know probability distribution for parameter $\phi$. I have the empirical distribution/statistical distribution of $X$ that is dependent on parameter $\phi$ for $\phi \in [0,1]$. I assimilate this ...
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### Probability distribution for transformation of a random variable

Let $g(x)=1$ if $x \leq c$ and $g(x)=(1-x)/(1-c)$, where $0 \leq x \leq 1$ and $0 <c <1$. So $g$ is an non-increasing function. Define $g^{-1}(y)=\inf\{0 \leq x \leq 1 \mid g(x) \leq y\}$. ...
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### Are the products of different independent random variables independent?

I'm confused about the independence of the product of independent random variables. Let A and B be independent of each other ...
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### How to show that polar coordinates in a uniform distribution on a disk are independent?

Let the random point $(X,Y)$ be uniformly distributed on the unit disc $D=\{(x,y):x^{2}+y^{2}<1\}$. Show that the polar coordinates $R\in [0,1)$ and $\theta \in [0,2\pi)$ of the point are ...