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

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6answers
212 views

Producing a CDF from a given PDF

So I have this PDF: $$ f(x)= \begin{cases} x + 3 & \text{ for } -3 \leq x < -2\\ 3 - x & \text{ for } 2 \leq x < 3\\ 0 & \text{ otherwise} \end{cases} $$ To make this a CDF, I ...
2
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2answers
65 views

Probability Distributions and Probability

Suppose $X \sim N(3, 4)$, and let $Y = X^2$. Find $\Pr(Y ≥ 12)$. What does $Y$ mean?
2
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3answers
98 views

distribution of infinite sum of $\sum (2x_n -1)/2^n$

$\{X_n\}\sim\mathrm{Bernoulli}(\frac {1}{2})$ $$Y=\sum_{n=0} ^{\infty} \frac {2X_n -1}{2^n}$$ Find the distribution of $Y$ $X_n$ are independent
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2answers
63 views

Homework help finding pdf's of y given pdf's of x - stuck

If anyone can give me the steps on how to find pdf$\,'$s of $y$ given $x$. Let X be a continuous random variable with probability density function given by $$ {\rm f}\left(x\right) ...
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1answer
55 views

Is squared Brownian Motion a gaussian process?

I am working at the following SP, given by $(X_t)_{t\geq0} = \alpha W_t^2+\beta t$ where $W_t$ is Brownian motion and $\alpha,\beta$ real. I managed to calculate mean and covariance function and now I ...
0
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1answer
86 views

Conditioning on independent coin tosses - general solution to brute force method?

Consider 10 independent tosses of a biased coin with a probability of heads, $p$. question (4d): find the probability there are 5 heads in first 8 tosses and 3 heads in last 5 tosses. I managed to ...
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1answer
41 views

Limiting distribution of $n(T_n-4p^3(1-p))$

I want to find the limiting distribution of a $n(T_n-4p^3(1-p))$, where $T_n=\displaystyle\frac{4(n-t)t(t-1)(t-2)}{n(n-1)(n-2)(n-3)}$ with $t=\sum X_i$ is the UMVUE of $4p^3(1-p)$ that I found, where ...
1
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2answers
142 views

When to use alternate parametrization of Gamma distribution?

In Loss Models, 4th ed., by Klugman et al., the following parametrization is given for the Gamma distribution: $$f(x) = \dfrac{(x/\theta)^{\alpha}e^{-x/\theta}}{x\Gamma(\alpha)}\text{.} $$ When ...
0
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1answer
55 views

Deducing F-distribution PDF

Let $V\sim \chi^2(n)$ and $W\sim \chi^2(m)$ indep. r.v. I want to find the PDF for $X=\frac{V/n}{W/m}$. For that I define $h(v,w)=(v,v/n\cdot m/w)=(v,x)$. So, $h^{-1}(v,x)=(v,\frac{v \cdot ...
1
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1answer
67 views

Expected value of normal distributed variable

I need to calculate the expected value of a modified normal distributed variable but i'm struggling. So maybe someone can help me. Suppose we've got a normal distributed variable $X \sim ...
0
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1answer
22 views

How to determine the distribution of $U:=(X,Y,Z)$?

I've got a question concerning the distribution of a multi dimensional random variable. I know that $X$ and $Y$ and $Z$ are each normal distributed with certain expectations and variances. ...
1
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1answer
50 views

Coin toss with dynamic probabilities

So, I got a repeated experiment with two outcomes, i.e. a coin toss, but the probabilities might change every toss and are independent. Typically, they might come in sequences of the same ...
0
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2answers
48 views

If $X,Y$ ~$U(0,1)$ what is the distribution of $Z=0.5x^{2}+0.5y^{2}$?

I have some trouble with it.. the question is: $X,Y$ uniformly distributed $U(0,1)$ than $\frac{1}{2}(x^2+y^2) $~$exp(1)$... I am not even sure it is correct.. I know that if $X,Y$~$N(0,1)$ than it is ...
0
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1answer
57 views

Differentiation involving determinant

This question has arisen by following the proof in the appendix of Louis Liporace's paper on maximum-likelihood estimation, where the paper concerns classes of probabilistic functions (elliptically ...
4
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2answers
137 views

Distribution related to brownian bridge

Let $B(t)$ be a Brownian Bridge and $U$ is uniformly distributed on $(0,1)$. I wish to know the distribution function $B(U)$. Is it possible? As we know, $B(t)\sim N(0,t(1-t))$. But, I haven't a clue ...
1
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1answer
42 views

Homework help with Standard Normal Distribution

I have a homework problem in which I'm not certain where to start: Let $X$ be a random variable with $N (0, 1)$ distribution. Show that $E(X^n) =\left\lbrace{\begin{array}{cc} 0 & \text{if $n$ ...
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6answers
70 views

Distribution of a binomial variable squared

If I know $X$ is a binomial random variable, how can I find the distribution of $X$ squared (I know that $P(Y=y=x^2) = p(X=x)$ but does this distribution have a standard name)? In particular, how can ...
1
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2answers
326 views

Finding the probability an equation has real roots.

If $Q$~UNIF$(0,3)$, find the probability that the roots of the equation $g(t)=0$ are real, where $g(t)=4t^2+4Qt+Q+2$. There was a similar question asked that I looked at, but I am still a little ...
1
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1answer
54 views

How to determine distribution

I hope you will be patient with the inarticulate question of a non-mathematician. It's hard to get an answer when you don't even know how to ask the question. But here goes... ...Actually, I have two ...
5
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1answer
120 views

Proof of Interesting Binomial Identity

In my work I've come across the interesting binomial identity $$ \sum_{n\geq k} \frac{\binom{n}{k}}{\binom{m-1}{k}} \frac{\binom{m-1}{n} \binom{i-m-1}{j-n-1}}{\binom{i-2}{j-1}} = ...
5
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2answers
4k views

Deriving Moment Generating Function of the Negative Binomial?

My textbook did the derivation for the binomial distribution, but omitted the derivations for the Negative Binomial Distribution. I know it is supposed to be similar to the Geometric, but it is not ...
3
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1answer
52 views

Interesting Problem - Computing CDF

A rv X is an exponential distribution with parameter 1 and Y is a uniform distribution between 0 and 1. X and Y are independent. Define Z = min {X, Y}. Compute the CDF of Z ? I really have no idea ...
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0answers
39 views

One double integral elated problem

The bit I am stuck is the limits in the double integral. I tried X from 0 to uy and Y from 0 to infinity, this is obviously incorrect. I just want to know the complete double integral in the order ...
3
votes
1answer
46 views

expected value and variance of the difference of number of people in a row.

I need to calculate the expected value and the variance of the following variable: $n$ people sit in a row, among them person 'a' and person 'b'. Define $X$ to be the amount of people between 'a' and ...
0
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1answer
62 views

Compute a conditional probability of normal random variable

Suppose $X, T$ are continuous random variables, and $X \sim \mathcal{N}(0, 1)$, $T$ have density function $f_T$. (But $X,T$ do not have joint density) Is there any way to compute the following ...
0
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1answer
62 views

Covariance of random variables with identical distribution.

Let $X_1,...,X_n$ be random variables with identical distribution, and for all $i=1,...,n$ $\mathrm{Var}(X_i)$ exist. 1. Show that the covariance between each two random variables exist. 2. Show that ...
1
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3answers
81 views

Let $U$ be $~U [0,1] $and let $Y = U^{\frac{1}{2}}$

Let $U$ be $\sim \mathcal{U}[0,1]$ and let $Y = U^{1/2}$. I'm having trouble finding the $E(Y)$. How do I go about doing this?
2
votes
2answers
29 views

probability-distribution that has its mode equal median

Could anyone tell me any asymmetric distribution whose mode=median? Thanks in advance.
1
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2answers
335 views

Math probability combination explanation

A group of four components is known to contain two defectives. An inspector tests the components one at a time until the two defectives are located. once she locates the two defectives, she stops ...
1
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1answer
104 views

Estimate arrival time of a ship given the average of the ships in a day in a Poisson Distribution

I'm working in a simulation of a Port where ships come to specific stations of the port. I already know that the average amount of ships is given by a Poisson distribution and the service time (On ...
1
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0answers
141 views

Convergence of probability density functions

Assume that a sequence of random variables, $(X_t)_{t\geq 0}$, converges in distribution to a random variable $X_0$, as $t\to 0$. Also assume that $X_t$ and $X_0$ have $C^{\infty}$-probability density ...
0
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1answer
147 views

binomial coefficient: maximum value

For $n\rightarrow \infty$ we consider $$f(p)=\sum_{j=c}^n {n\choose j} p^j (1-p)^{n-j}.$$ We are interested in $\hat{p}:=\arg \max_p f(p)$. Can we say something about $\hat{p}$ dependent on $n$ and ...
8
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2answers
152 views

Lies, damned lies, and statistics

A story currently in the U.S. news is that an organization has (in)conveniently had several specific hard disk drives fail within the same short period of time. The question is what is the likelihood ...
1
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2answers
129 views

Does a continuous probability density function (pdf) have zero values on +infinity and -infinity?

Assume a pdf $f(x)$ is continuous along $-\infty$ to $+\infty$. Does this assumption guarantee that $f(+\infty)=f(-\infty)=0$? How to prove? Thanks in advance.
0
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1answer
41 views

monotonicity of binomial coefficient

I am interested in $$f(x):={k-1 \choose x-1} p^{x} (1-p)^{k-x}.$$ How do I find out in which Domain this function is monotonically increasing, in which it is monotonically decreasing? For which $x$ ...
1
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1answer
36 views

Cumulative Distribution Function and

The demand, $X$, for a firm’s product is a random variable with density $f(x) = 2x$ for $0 ≤ x ≤ 1$. The corresponding cumulative distribution function is $F (x) = x^2$ for $0 ≤ x ≤ 1$. The firm’s ...
0
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1answer
146 views

P(X>Y) Probability Double Integral

$f(x,y) = \frac{12}{7(x^2 + xy)}$ $ 0 \le x \le 1$ and $0 \le y \le 1 $ I want to know the $P(X>Y)$. I believe the correct solution to this is integrating from 0 to 1 for dy and y to 1 for dx ...
1
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0answers
74 views

Sums of Power Law random variables

Suppose $F$ is a Pareto distribution with scale parameter $x_m$ and shape parameter $\alpha$. Assume $X_1, X_2 , \dots, X_n$ are iid random variables drawn from $F$. Let $S_n(k) = X_1 ^k + X_2 ^k + ...
0
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1answer
95 views

An exercise on quantile from Michael Wichura's notes

Please help me with this (source and context follows after the question). Thank you! Question: Let $F_1,\ldots,F_n,\ldots$ and $F$ be distribution functions with corresponding quantiles ...
3
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1answer
589 views

Precise definition of the support of a random variable

I am reading lecture notes which contradict my understanding of random variables. Suppose we have a probability space $(\Omega, \mathcal{F}, Pr)$, where $\Omega$ is the set of outcomes ...
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0answers
44 views

Probabilistic fragmentation

Suppose we have the following problem: We start with an interval of length $1$ and break it into two intervals of lengths $r$ and $1-r$, where $r$ is a random variable in $[0,1]$ with probability ...
1
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1answer
91 views

If $X,Y$ are independent and geometric, then $Z=\min(X,Y)$ is also geometric

Let $X,Y$ be independent geometric random variables with parameters $\lambda$ and $\mu$. If $Z=\min(X,Y)$. Show that $Z$ is geometric and find its parameter. (Answer $\lambda\mu$) $\displaystyle ...
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1answer
135 views

Correlation coefficient of i.i.d variables

Let $X_1, X_2, X_3, ...$ be i.i.d variables, and for every $i$ $X_i$ has variance. Define $S_k=\sum_{i=1}^{k}X_i$. Calculate $\rho(S_m,S_n)$ for $m\leq n$. Well, I know it should be $\sqrt{ m/n }$, ...
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1answer
45 views

What are the different ways of indicating that a random variable has a specific distribution?

Recently I have seen random variable distributions described in two ways: $$ X \sim Nb(r,p) \\ X \stackrel{d}{=} Nb(r,p) $$ Both indicating that $X$ is a negative binomial random variable with $r$ ...
0
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1answer
27 views

Probability “average” understanding

This is more of a problem understanding probabilities than an actual question. In a game I am playing I can use a certain item to try to unlock different levels. The item will unlock a new level ...
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0answers
58 views

Extension of Slutsky's Theorem

I regard random variables $X_n$ and $Y_n$ with $(X_n+Y_n) \rightarrow (X+Y)$ (in distribution for $n \to \infty$). Furthermore there exist random variables $(a_n) \rightarrow 1$ and $(b_n) \rightarrow ...
1
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0answers
73 views

Conditioning on function of random variable and random variable itself

Suppose that $Y_{i}\in\{0,1\}$ is a binary variable, and $X_{i}$ is some random vector in $\mathbb{R}^{d}$ . Why can we say the following: \begin{eqnarray*} ...
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1answer
42 views

Confused how to calculate continous random variable with pdf that has a min

The problem given was: Let $X$ be a continuous random variable with probability density function $$f(x) = \dfrac 1 4 \min \left( 1, \dfrac 1 {x^2} \right)$$ Find $P(−2 \le X \le 4)$. The ...
1
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1answer
40 views

Finding Probabilities of Distribution Functions

I recently turned in an assignment and had an error on it, or so I'm told, I'm not entirely convinced just yet. The problem was as follows: $$F(x) =\begin{cases}1-\frac{16}{x^2}, & x\ge4 \\ 0, ...
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0answers
125 views

What does “sequence is equidistributed in [0, 2]” mean?

I was reading an article in which they are mentioning this sentence: "sequence is equidistributed in [0, 2]" where the sequence in question, is a sequence of real number (the article in question is ...