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

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2answers
27 views

What is the joint distribution of these two obscured exponential ones?

$X$ and $Y$ are independent random variables with $X \sim exponential(\lambda)$ and $Y \sim exponential(\mu)$. It is impossible to obtain direct observations of $X$ and $Y$. Instead, we observe the ...
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1answer
10 views

pdf (transformations of variables

If X has the pdf $f(x)=\frac13, -1<x<2$, zero elsewhere,find the pdf of $Y=X^4$. here is my solution: The support of $Y$ is $(1,16)$. Now, $P(Y\le y)=P (X\le y^{\frac14})$.. then the cdf of Y is ...
2
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1answer
44 views

Find upper limit of normal distribution integration

Considering the normal distribution with standard deviation equals to 0.9 and mean 2.1: $$ P(X\leq a) = \frac{1}{0.9\sqrt{2\pi}}\int_{-\infty}^{a} e^{-\frac12\frac{(x-2.1)^2}{0.9^2}}\,dx $$ I must ...
2
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2answers
22 views

CDF of independent variables

I am given two independent variables $X$ and $Y$. Where $F_X(x)=F_Y(x)=x^4 \ 0\le x \le 1$. I am looking for CDF of $Z=max(X,Y)$ and $E(Z^2)$ I need some pointers, especially for $Z^2$. Thank you.
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1answer
26 views

Need to find the distribution density of a random vector [closed]

I have two independent variables $X$ and $Y$ with distribution functions $$f_X(x)=x, \ f_Y(x)=x$$ such that $0\le|x|\le1$. I need to find the distribution density of a random vector ...
4
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0answers
35 views

Distribution of points in ellliptic curves over finite fileds

Let $E$ be an elliptic curve defined over a finite field ${\bf F}_p,$ where $p$ is prime. From Hasse theorem we get $p+1-2\sqrt{p} \leq |E({\bf F}_p)|\leq p+1+2\sqrt{p}.$ Now say that we choose in ...
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1answer
26 views

How to calculate the sum of that probability distribution? [duplicate]

Let $W$ be the random variable that counts the number of tails before one gets $r$ heads for a coin whose probability of heads is $θ$. Without using moment generating function, show that the mean and ...
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4answers
49 views

about a difficult and weird Probability question

Let W be the random variable that counts the number of tails before one gets r heads for a coin whose probability of heads is θ. Without using moment generating function, show that the mean and ...
0
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1answer
44 views

conditional probability that 5 red balls were placed in the bowl at random

Place five similar balls (each either red or blue) in a bowl at random as follows: A coin is flipped 5 independent times ad a red ball is placed in the bowl for each head and a blue ball for each ...
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0answers
19 views

Testing if distribution is geometric and finding is parameter.

I was doubting whether this was better suited on a math page or on a programming page but I thought just start with the fundamentals. I'm working on the following: I have constructed a exponential ...
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0answers
21 views

Probability of this event of a tuple not being deleted: Is this correct?

We consider a set of n tuples. Now, among the $2n$ elements, $m$ are deleted uniformly at random. Now the question is, what is the probability that 1 specific tuple is not deleted. I thought it ...
6
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1answer
81 views

How Do I Find My Car

I have been discussing this problem with a coworker for a few days now and neither of us have made any headway on it. I would appreciate any help with a possible solution or maybe a suggestion of a ...
1
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0answers
16 views

Expectation of a function of Gamma random variable

Consider a truncated exponential distribution $F(x\left| \lambda \right.) = \frac{{ - {e^{ - \lambda x}} + {e^{ - \lambda }}}}{{ - {e^{ - 2x}} + {e^{ - \lambda }}}}$ on the interval $[1,2]$. The ...
1
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1answer
20 views

Sum of normally distibuted random variables?

I feel terribly confused over this. If: X~N(μx,σx^2) Y~N(μy,σy^2) both independent random variables Z=X+Y Z~N(μx+μy,σx^2+σy^2) Then why: Xi~N(μi,σi^2) i=1, 2, ..., n X1, X2, ..., Xn are ...
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1answer
16 views

Domain in marginal density functions

I am so confused about the domain for marginal density of this problem... Here is the joint density function : Here is the solution: I perfectly understand how we get those two marginal ...
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0answers
22 views

Marginal pdf from joint

I'm not sure I understand something correctly; I have 3 random variables, $S_1, S_2, S_3 $ and I know their joint pdf, $f_{S_1, S_2, S_3}(s_1, s_2, s_3). $ The domain for my random variables is ...
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0answers
7 views

Multivariate convolution density?

In reference to this post, the pdf for dependent random variables $X_1+X_2$ is given by: $$f_{X_1+X_2}(z) = \int_{-\infty}^{\infty} f_{X_1,X_2}(x,z-x) \mathrm dx$$ How does this formula extend to ...
2
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1answer
32 views

How to use expectation of a random variable to prove its distribution?

If we know a random variable $X$ satisfies $$E(X^k) = \left\{ \begin{array}{lll} 1 \cdot 3\cdot ...\cdot k & \text{if} & k \text{ is even}\\ 0 & \text{if} & k \text{ is odd}\\ ...
1
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0answers
35 views

Product distribution function of two independent random Variables

Why, if $X $ and $Y $ are two independent'', continuous random variables, described by probability density functions $f_X $ and $f_Y $, then the distribution of $Z = XY$ is $$f_Z(z) = ...
1
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1answer
36 views

Variance of number of cycles of length $t$ in a permutation

We consider a uniform Distribution over all $n!$ permutations of $\{1, \dotsc, n\}$. Now we are interested in the Variance of the number $C$ of cycles of length $t$. We have $$E[C]={n \choose t} ...
2
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1answer
20 views

Product of a Rademacher and a standard normal random variable

$X \sim N(0,1)$ and $Z \sim Rademacher$, and they are independent. How can I show formally that $Y=XZ \sim N(0,1) $ too?
2
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1answer
173 views

Probability question involving simulations of picking balls from a bag

I’m working on a chemistry problem, which essentially translates to finding the answer to a related probability problem. However, my knowledge in probability is very limited and I'd be grateful if ...
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1answer
12 views

PDF of uniform distribution

Find the uniform distribution of the continuous type that has he same mean and the same variance as those pf a chi--square distribution with 8 degrees of freedom. My solution: for the mean- $\frac ...
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3answers
167 views

Distribution and second order differential inequality

I would like to solve the following $2^{\mathrm{nd}}$ order differential inequality $$ \theta_F'(x) = \frac{2F'(x)^2 - F(x)F''(x) + F''(x)}{F'(x)^2} < 0 $$ for some subinterval $I \subset ...
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1answer
27 views

Convergence In probablility implies convergence in distribution?

I'm currently working on the following exercise: If $X_n$ is sequence of randon maps with values on a metric space $(S,d)$. Show tha convergence in probablity to a randon map $X$ implies $P\circ ...
3
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3answers
159 views

A specific kind of probabilistic proof for central binomial coefficients

I'm looking for a specific kind of proof of the statement $$ \lim_{n\to\infty} \frac1{4^n}\binom{2n}{n} = 0 $$ I know how to show this using Stirling's formula; I have seen the very nice elementary ...
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1answer
12 views

Checking the convergence in distribution

Random variables $X_1, X_2, ...$ are independent and for every $n$ the random variable $X_n$ is uniformly distributed on the interval $[-\sqrt{n}, \sqrt{n}]$. Denote $\sigma_n^2 = \sum_{k=1}^{n} ...
0
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0answers
13 views

Chernoff Bounds for non-Bernoulli random variables

Normally, one applies the Chernoff bound for a sum of independent Bernoulli random variable. But what about if there are other outcomes? For a sum of random variables with outcomes $0, c$, one can ...
1
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1answer
26 views

Expectation of multinomial distribution

Three fair dice are cast. In 10 independent casts, let X be the number of times all three faces are alike and let Y be the number of times only two faces are alike. Find the joint pdf of X and Y and ...
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2answers
24 views

Smallest value of n

Let $X$ have a binomial distribution with parameters $n$ and $p=1/3$. determine the smallest integer n can be such that $P(X\ge1)\ge0.85$. In this problem I am stuck with $\frac{2^n ...
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1answer
24 views

Probability of unbiased die

One of the numbers 1,2,...,6 is to be chosen by casting an unbiased die.Let this random experiment be repeated five independent times.Let this random variable $X_1$ be the number of termination in the ...
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0answers
12 views

How are these distributions called?

Are these any common distributions? Number of dependent trials until the first success occurs Inspired by (a) in this question here: Probability Problem with $n$ keys Binomial-like distribution ...
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0answers
2 views

Conditions on distributions to obey a certain inequality

Given a random variable X with (continuous, differentiable) CDF $F(x), x\geq 0$, I want to find conditions under which it satisfies ($\forall a,x\geq 0$): ...
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1answer
10 views

Fisher exact text and connection between Binomial and Hypergeometric distributions.

My textbook shows the connection between binomial and hypergeometric using the fisher exact test.."Assuming the null hypothesis and letting p=p1=p2, we have $X$ ~ $Bin(n,p)$ and $Y$ ~ $Bin(m,p)$, ...
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1answer
19 views

Joint PDF Correlation

In the problem I am given $f(x,y)=2,\ 0 < x < y,\ 0 < y <1$. I'm trying to find the correlation $\rho$ which I know is equal to $$\rho = \frac{Cov(x,y)}{\sqrt{Var(x)Var(y)}}$$ ...
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0answers
8 views

Test for independent and stationary of time series

I am trying to do some basic time series analysis. Suppose that I have a real data time series, how would I test for independent and stationary increments? Regarding to stationary, I am thinking of ...
4
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2answers
49 views

How do you find $f(x_1, x_3)$?

$X_i$ is the number of times (out of 100) that a die's face has $i$ dots. I know that $X_i\sim \text{binomial}(100, 1/6)$, so $f(x_i)={100 \choose x_i}(1/6)^{x_i}(5/6)^{100-x_i}$. How do you find ...
0
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1answer
16 views

Probability beta distribution problem

A beam of length $1$ is rigidly supported at both ends. Experience shows that whenever the beam is hit at a random point, it breaks at a position $X$ units from the right end, where X is a beta random ...
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1answer
47 views

The probability of hitting a bulleye

Lisa shoots at a target. The probability of a hit in each shot is 1 /2. Given a hit, the probability of a bull’s-eye is p. She shoots until she misses the target. Let X be the total number of ...
3
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0answers
66 views

When will this generalized binomial model generate an exchangeable sequence?

Start with a generalized binomial model $$P(X_{n+1}=1\mid \mathcal{F}_n)=\theta_n+ n^{-1} d_n \sum_{i=1}^n X_i$$ $$P(X_{n+1}=1)=p_{n+1}=\theta_n + n^{-1}d_n \sum_{i=1}^n p_i$$ $$P(X_1 = 1)= ...
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1answer
23 views

Beta-binomial random number generator

Could someone help me find a random number generator from a Beta-Binomial distribution in MATLAB, R or SAS? Thank you!
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1answer
39 views

density joint function

I got a question and I was stuck for more than 15 minutes... Here is the question, And the question was: Find F(1/2,2). I tried to reason but the answer was different from what I got, here is the ...
3
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1answer
26 views

Series of independent gaussian variables and brownian motion

I am checking the proof of the construction of a brownian motion in $[0,\pi]$. We show that \begin{gather*} t \mapsto B^m_t = \frac{t}{\sqrt{\pi}}X_0 + \sqrt{\frac{2}{\pi}}\sum_{n=1}^{2^m-1}X_n ...
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0answers
38 views

Generalized Binomial Model independent in the limit

Start with a generalized binomial model $$P(X_{n+1}=1\mid \mathcal{F}_n)=\theta_n+ n^{-1} d_n \sum_{i=1}^n X_i$$ $$P(X_{n+1}=1)=p_{n+1}=\theta_n + n^{-1}d_n \sum_{i=1}^n p_i$$ With $0\leq \theta_n+ ...
3
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0answers
26 views

Probability density function of $x$ in the unit circle?

I'm trying to work out how to find the probability density function (PDF) for $x$ values on the unit circle - not within the unit circle but on the edge. The reason for doing so is that I'm trying to ...
0
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2answers
92 views

Express the CDF of $Y=X^2$ [closed]

Let $X$ be a random variable with CDF $F$. Express the CDF of $Y=X^2$ in terms of $F$.
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1answer
48 views

Conditional Distribution: how to set up Limit of Integration of a joint density

I have a question in conditional probability. I'm asked to find the conditional distribution, however, I'm unsure about the answer given and would appreciate someone helping straighten out the theory ...
0
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0answers
47 views

How to develop a probability distribution/density function of an issue?

Assume that a health insurance company has $1000$ customers. It is estimated that the probability of a customer making a claim is $p = 0.2$ per year, independently of previous claims and other ...
0
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1answer
37 views

Random variable related to binomial

The number of successes $A$ in $n$ independent trials with the probability of a success is $p$ for each trial is binomially-distributed. I am interested in a scenario that adds dependence to the ...
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2answers
77 views

What is the distribution of Z=min(X,Y) [closed]

Let X and Y be independent geometric random variables. What is the distribution of Z=min(X,Y)?