Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.

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2answers
15 views

Is multiplication normally/binomially distributed?

I was thinking about the binomial formula in the context of coin flips and got to thinking about the reason that even though HHHHHHHHHH is just as likely to occur as a sequence as HHHHHTTTTT, 5 heads ...
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0answers
8 views

What is the confidence level of a curve fitted moving average forecasting next element in time series?

Suppose I got a time series X={X1,X2,X3,...,Xn}, and i got a Moving Average of any kind, it doesn't matter. The moving average is curve fitted to the timeseries of n elements, to fit exactly and most ...
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0answers
18 views

Multivariate Probabilities

Let the random variables X and Y have the joint probability density function given by: $f(x,y)=96/7xy$ if $0<x<1$, $x$-1$<$y$<1-$x\2 f(x,y)=0 if otherwise (a) Find the marginal ...
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0answers
16 views

A minimization question about the convexity of KL-divergence

Let $f_1$ be a continuous density function which is given and consider the closed ball around $f_1$: $$\mathcal{G}=\left\{g:\int g(x) \ln\frac{g(x)}{f_1(x)}\mathrm{d}x \leq \epsilon\right\}$$ ...
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1answer
8 views

Bernoulli trials case in probability

A fair die is tossed twice. About how many times would you expect to roll 3 or greater? So based on sequence of Bernoulli trials: P(exactly k successes in n trials) = C(n,k) p^k q^(n-k) where p = ...
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0answers
25 views

probability distribution function of two independent variables

Let $X$ be a random variable whose distribution function is $F_X(t)=3^{-t}$. Suppose that $Y$ is another random variable whose distribution function is $F_Y(t)=4^{-t}$. What is the probability that at ...
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3answers
31 views

Expected number of rolls on a dice?

You roll a die until you have seen a 5 on 4 of the rolls (e.g. ⟨5,3,2,5,4,1,6,5,2,5⟩. What is the expected number of rolls this will take? I think that I am way overthinking how I should be going ...
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3answers
14 views

Probability normal distribution P(X>Y)?

Angelo earns every month as a variable normal X N(1000;400^2), Bruno N(1400;300^2). Calculate the probability of Angelo earns more then Bruno p(X>y)?
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1answer
24 views

Expectation over 2 random variables, help needed

Hi I am new here and I hope I can get some help. My question is about taking expectation over random variables. Lets say I have two random variables $\Xi$ and $\theta$ where $\Xi$ is for example a ...
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0answers
17 views

Urn problem (possibly a coupon collectors problem)

In an urn with 10 different coloured balls (each colour has an equal chance to be selected, let's say m balls of each colour). Can I find the mean number of draws to : Have one colour from 10 ...
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2answers
23 views

How to find distribution of kth highest of n iid's?

For example, suppose there are five people whose heights are independent and exponentially distributed with mean 5.5 feet. I want to be able to solve problems like 1) Find the expected height of the ...
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2answers
19 views

Probability distribution over time of an event dependent on a prior event

Suppose I have an event $A$ that can only occur once in a experiment. A large ensemble of experiments reveals that $A$ occurs at a rate $r\, \mathrm{d}t$. For simplicity take $r$ as a constant. The ...
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1answer
10 views

Probability that two points (any where on the curve) are a set number of standard deviations apart on a normal distribution

So, here is the question: You buy two pieces of pipe from supplier A, and the inner diameter has a normal distribution of N(muA, sigmaA^2) = N(8.02, 0.1^2). You want these two pipes to butt together ...
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0answers
10 views

Matching Gamma Statistics with Poison Statistics

Confidence intervals with Poisson distribution would be greatly helped by matching an equivalent gamma distribution. Can someone lay out how to match a Gamma Distribution to a poisson distribution? ...
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0answers
12 views

Sampling from the Marshall-Olkin Copula

I know the algorithm, just cannot understand why it works. There are 2 processes with survival times $T_1,T_2$, and we wish to compute the probability that they survive shock times $t_1,t_2$. $$ ...
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0answers
23 views

Relation between minimum and sum of two random variable

I am interested in finding a relation that involves two independant random variables, that could be used to describe the sum of these, or the minimum of these. For example, regarding the sum, we know ...
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1answer
23 views

Maximum of Correlated random variables

I am trying to find the CDF $Z = \max(X_1,X_2,\dots,X_N)$ and in my case $X_i$ are correlated. Is there any transfer domain or one to one function where I can derive the CDF and invert back to current ...
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1answer
13 views

Equality of two binomial parameters

I am having some trouble with this problem of Ross. Can anyone please help me out. In a famous experiment to determine the efficacy of aspirin in preventing heart attacks, 22,000 healthy middle-aged ...
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0answers
17 views

How to solve using Jacobian?

I am looking for a way to find the joint pdf of vector $Z=[Z_1,Z_2,Z_3,Z_4]$ where $Z_1= a_1 X_1^2 + a_2X_1Y_1+ a_3 X_1Y_2 + a_4Y_1^2 + a_5Y_2^2$ $Z_2= b_1 X_1^2 + b_2X_1Y_1+ b_3 X_1Y_2 + b_4Y_1^2 + ...
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0answers
23 views

A reference for a Gaussian inequality ($\mathbb{E} \max_i X_i$)

I am looking for a reference to cite, for the following "folklore" asymptotic behaviour of the maximum of $n$ independent Gaussian real-valued random variables $X_1,\dots, X_n\sim ...
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1answer
18 views

Given a data set of random values in MATLAB, how do I plot its PDF?

Suppose I have the following code in MATLAB: ...
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1answer
23 views

uniqueness of joint probability mass function given the marginals and the covariance

Let X and Y be two nonnegative, integer-valued random variables. Is there a way to find the joint probability mass function, i.e. $$ \mathbb{P}(X= k, Y= h) $$ for some $k,h\geq 0$, given the ...
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1answer
29 views

How many will not be selected in repeated tries?

Suppose I have $25$ uniquely identifiable objects, i.e., I know which is which once it has been selected (but they are not distinguishable in the selection process). I select $5$ objects at random, ...
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0answers
20 views

Find $E[Z_1 | aZ_1 + bZ_2]$

Let's $Z_1,Z_2$ be a random variable such that $EZ_1^2 < \infty$ and $EZ_2^2 < \infty$. Find $E[Z_1 | aZ_1 + bZ_2]$ where $a,b \in \mathbb{R}$. We don't know what is distribution of $Z_1$ and ...
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1answer
34 views

Conditional probability problem and Alias Method

I hopefully someone can help me with this problem of conditional probability: "A disk server receives requests from many client machines and requires 10 milliseconds to respond to each request. The ...
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0answers
28 views

how to related a weakly convergent random variable with its k-th moment

Let $\{X_n\}$ be independent random sequence with zero mean and unit variance. Suppose $$S_n:=\sum_{m=1}^n \frac{X_m}{\sqrt{n}} \Rightarrow X\sim \mathcal{N}(0,1)$$ holds. (Here "$\Rightarrow$" ...
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0answers
37 views

necessary conditions for these conditionals to be consistent with some joint distribution

Let $A$, $B$, and $C$ be random variables taking discrete values in the set $\{0,1\}$. I'm trying to find necessary conditions such that the conditional distributions $$X\mid Y,\,Y\mid Z,\,Z\mid X$$ ...
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0answers
16 views

Probability distribtuions [on hold]

A 10 metre by 10 metre plot of land is divided into 100 equally sized squares. Suppose that 300 seeds are randomly scattered on the plot of land. Use a suitable approximation to find the probability ...
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0answers
19 views

Cameron Martin Theorem

I am struggling with two versions of the Cameron Martin Theorem. 1) We define the measure spaces $(\Omega,\mathcal{F},P)$ and $(C[0,1],\mathcal{C},\mathbb{L}_0)$, where $\mathcal{C}:=\sigma(f\mapsto ...
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2answers
15 views

Finding a constant from a continuous distribution

$X$ is a continuous random variable with PDF $$f(x) = c\theta^{|x|} \quad \text{ for } -\infty<x<\infty,$$ where $0<\theta<1$ is a parameter and $c$ is a constant. Derive and expression ...
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0answers
17 views

Function of Nakagami Distribution

Does anyone know what the distribution of the sum of squared Nakagami is? $$\sum_i^n X_i^2$$ $$X_i\sim \text{Complex Nakagami-m }$$ Is the distribution Erlang? Is the distribution the same as ...
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1answer
8 views

How do you compute the PDF of a function of 2 random variables that is not a sum?

If you have a random variable U(X,Y) that is a function of two other random variables X and Y such that $U(X,Y)=X+Y$ and you know the PDFs of X and Y are defined to be exponential such that $f(t) ...
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1answer
9 views

How to interpret this variance

If I have a probability measure defined by $P( \Omega ) = \int_{\Omega} (1-a) \delta(x) + a \delta(x-a^2) dx,$ then I noticed that the variance is given by $a^5(1-a)$. This is somewhat strange, cause ...
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0answers
19 views

how to find expected value with toys? [duplicate]

A couple years ago, Burger King was giving a Dragon Ball Z toy in every kids meal. There were 6 unique toys that you could collect. Lets say I randomly selected a toy in each kids meal. What is the ...
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0answers
17 views

What is the magnitude of Complex random variable Gaussian Case?

Let $X_1$ and $X_2$ be independent complex Gaussian random variables, $$X_1 \sim \mathcal{CN}(0,\sigma)$$ $$X_2 \sim \mathcal{CN}(0,\sigma)$$ If $X= aX_1 + bX_2$ where $a,b$ are constants then the ...
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1answer
25 views

marginal pdf of a exponential distribution

Problem Let $X$ have the pdf $f(x)=e^{-x}$, $x>0$ and $Y$ have the pdf $f(y)=e^{-y}$, $y>0$. Assume that X and Y are independent Find the pdf of $U=X+Y$? Solution Since X and Y are indep. it ...
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2answers
30 views

For a non-negative absolutely continuous random variable $X$, with distribution $F$. Why is $\lim_{t\rightarrow \infty}t(1-F(t))=0$?

So I am given a non-negative absolutely continuous random variable $X$ with distribution $F$, and density $p_X$. I am given the definition of expectation using simple functions and the survival ...
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0answers
18 views

statistics uniform distribution [on hold]

Dominic released his rabbit to roam on the lawn; some time later, it returned and so he continued doing that daily. Over time, he found a pattern of the time T (in hours) Rabbit stayed outside. If ...
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1answer
24 views

A question of joint CDF

I am confused about how to use the joint probabilities to find the joint CDF.
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0answers
12 views

Probability that one random variable is greater than or equal to another

Assume $X$ and $Y$ are i.i.d. with exponential distribution with parameter $\lambda = 1$ (the probability density functions $p_X (x) = e^{-x}$ and $p_Y (y) = e^{-x}$ in $[0, +\infty)$, $0$ otherwise). ...
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0answers
2 views

Unable to follow notation and meaning of probability distribution for binary time series

I am unable to understand concepts related to the probability distribution of binary time series. [Mathematics is not my background]. This is from the book Binary time series by Benjamin Kedem, vol ...
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1answer
14 views

Probability of Normal Distribution

Let's say that 10 sumo wrestlers were to squeeze into an elevator that could only hold a max capacity of 2300 pounds. Let's say that the weight of the sumo wrestlers is normally distributed with a ...
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1answer
20 views

Elementary Probability: Expected Value

I must say, first, that this question IS a homework assignment and I do not wish an answer here, for I already posssess it. I want to know if there is a general procedure of simplification in this ...
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1answer
19 views

A question of Joint PDF

I have not idea about part a. I know I need to prove the integration of f(x,y)=1, but how should I deal with the range of x and y.
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0answers
11 views

Deriving a simple PDF

I am looking for deriving the pdf of $Z$ where $Z= (\sum\limits_{i=1}^N a_i X_i +Y_1)^2 + (\sum\limits_{i=1}^N b_i X_i +Y_2)^2$, where $X_i$ and $Y_i$ are independent, zero mean Gaussian random ...
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1answer
28 views

Cumulative distribution function of exponentials

I have the cumulative distribution function $F(x)=(1-e^{-x})\mathbb{1}_{x≥0}$ and want to write the CDF to $F(\frac{x-\mu}{\sigma})$. I have derived ...
0
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1answer
16 views

Conditional Expectation of Binomial Given $X \leq x$

Are there any neat formulas to reduce something like $\sum_{i=0}^{x} i \binom{n}{i} p^i (1-p)^{n-i}$ where $x<n$? This would be proportional to $\mathbb{E}(X\leq x)$ where $X$~$\text{Bin}(n,p)$. ...
0
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1answer
13 views

Joint CDF from conditional cdf

I would like to derive an expression of the following joint CDF $P[X \leq x,Y \leq y]$ based on the conditional CDF $P[X \leq x | Y=y]$ and the pdf $P[Y=y]$ that are considered to be known. I get a ...
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1answer
33 views

Is $r_2$ a uniformly at random value in $Z_n$, where $r_2=r_1 . m$

Let $m$ be an arbitrary value in $Z_n$, where n is RSA modulo (n=p.q, where p and q are large primes). Then have: $r_2=r_1 . m$, where $r_1$ is a value chosen uniformly at random : $r_1\in Z^*_n$. ...
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0answers
24 views

The Joint PDF question [on hold]

Can someone help me do this series of question? I really need help and I have no idea about it. I have no idea of how to deal with the range of x and y.