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

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2answers
2k views

Probability of one normdist being greater than another [duplicate]

I have two independant normally distributed random variables. X ~ N(657, 3) Y ~ N(661, 2) P(x > y) = ? How do I calculate the probability of X being greater ...
0
votes
1answer
373 views

covariance of two linear combinations of a bivariate normal distribution

$X$ and $Y$ are jointly normal, with the mean vector and covariance matrix given by: $$\mu= \begin{pmatrix} 1 \\ 2 \\ \end{pmatrix} \Sigma= \begin{pmatrix} 2 & 0.4 \\ 0.4 & 1 \\ ...
1
vote
0answers
204 views

To obtain the closed-form expression of CDF and PDF from the recurrence relation

Now I have a question, in which I need to find the probability mass function and the cumulative distribution function. But now I only have the recurrence relation. Here is the details: Assume ...
1
vote
1answer
802 views

Approximating a sum of two binomial distributions

A club basketball team will play a 60-game season. Thirty-two of these games are against class A teams, and 28 are against class B teams. The outcomes of all the games are independent. The team will ...
1
vote
1answer
83 views

biasedness/unbiasedness of an MLE.

To show whether an MLE I just found is biased/unbiased, would I need to find the expectation of the answer? Plus would I do this by integrating $\text{MLE} \cdot \text{pdf}$. My MLE is $ ...
0
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1answer
73 views

CDF for random variable $X(\omega) = 2(1-|2\omega - 1|)$

I don't know how to calculate this cdf, the modulus is very annoying, because the cdf definition is $P(X< x)$ in my case $P(\omega < x)$. But in the modulus equality I get this $P(-\omega < ...
2
votes
2answers
2k views

How do I read this distribution function: $\min(X,Y)$?

I'm confused on what the $\min$ means. For example if I need to find the distribution function of $\min(X,Y)$ what am I looking for exactly? Am I looking for the distribution of the minimum value of ...
1
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2answers
74 views

Stats - Likelihood function

Let $X_1, X_2, \ldots , X_n$ be a random sample from a distribution with the following pdf $$f(x|\theta) = \begin{cases} 1/(\theta_2−\theta_1), &\quad\text {for}\quad \theta_1 \leq x\leq ...
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1answer
2k views

How to calculate sum of negative binomial distribution ? it E[X] Var[x] and P(X = n |1 st event occurs on the 5 th try)

How to calculate sum of negative binomial distribution ? it $ E[X], Var[x]$ and $P\{X = n |\text {first event occurs on the 5-th try}\}$ $$P(X=k)= \binom{k-1}{r-1}p^r(1-p)^{k-r}$$
2
votes
2answers
166 views

Why is $\frac1\pi$ the answer for this Cauchy Distribution?

So I have this problem here, and I know what the correct answer is but I'm not sure of why it's the right answer. It says: Let $X$ be a random variable with the density function $$f(x)= \frac{1}{\pi ...
0
votes
1answer
124 views

How do I show that $P(N=n) = \frac{x}{n} P(X=x)$

Suppose that independent Bernoulli trials with parameter $p$ are performed successively. Let $N$ be the number of trials needed to get $x$ successes, and $X$ be the number of successes in the first ...
1
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1answer
153 views

Probability question on conditional prob

Suppose a-priori chance of getting malaria is 10%. A positive blood test indicates a 80% chance of actually having the disease; but 5% of time healthy people also test positive. Suppose you test ...
1
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2answers
83 views

Find the distribution of $X$ given that $P(X=k)=\frac 23 (k+1)P(X=k+1)$

A discrete random variable $X$ of values in $\mathbb N$ verifies the property that $$P(X=k)=\cfrac 23 (k+1)P(X=k+1)$$ What is the distribution of $X$? I found that $$P(X\ge 0)=\sum_{k=0}^\infty ...
0
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2answers
65 views

If $\text P(X>k+1)=\cfrac 12 \text P(X>k)$, what is the distribution of $X$?

If $\text P(X>k+1)=\cfrac 12 \text P(X>k)$, $k\in \mathbb N^*$ what is the distribution of $X$? So I did; $$\text P(X=k+1)= \text P(X>k)- \text P(X>k+1)=\cfrac 12 \text P(X>k)$$ ...
0
votes
1answer
166 views

Product of two independent stochastic variables $XY$

I have two independent variables $X\sim \mathcal B(n,p)$, Binomial and $Y\sim \mathcal P(\lambda)$, Poisson. How would I go about finding the distribution of $Z=XY$ and the couple $(Z,S)$, where ...
1
vote
2answers
303 views

Proof of $\sum_{k=0}^n k \text{Pr}(X=k) = \sum^{n-1}_{k=0} \text{Pr}(X>k) -n \text{Pr}(X>n)$

$X$ is a random variable defined in $\mathbb N$. How can I prove that for all $n\in \mathbb N$? $ \text E(X) =\sum_{k=0}^n k \text{Pr}(X=k) = \sum^{n-1}_{k=0} \text{Pr}(X>k) -n ...
1
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1answer
1k views

How to prove/show that some distribution is symmetric at 0

I have to prove that the Cauchy distribution is symmetric at 0. However, I'm not entirely sure how to do this. I'm given the problem: Suppose that a particle is fired from the origin in the $(x,y)$ ...
1
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0answers
137 views

generating a binomial distribution

I'm trying to sample from a data set using a binomial distribution with parameters p and n. Implementation-wise, I follow these steps I generate an array containing the values of the cumulative ...
0
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2answers
261 views

M/GI/1 service time distribution

I want to compute the distribution of the waiting time and the number of jobs for M/GI/1 where the service time is Heavy-Tailed or more specifically Pareto. I found this paper ...
3
votes
1answer
106 views

A question about infinities and distribution functions

Let $\mathcal{P}_i$ be the set of probability density functions to which $f_i$ belongs, $(i=0,1)$. Furthermore assume that $$L(y)=\frac{f_1(y)}{f_0(y)}$$ is an increasing function for any chosen ...
0
votes
0answers
396 views

pmf of X: the number of defects per yard

The number of defects per yard, denoted by X, for a certain fabric is known to have a Poisson distribution with parameter $\lambda$. However, $\lambda$ is not known, and is itself assumed to be random ...
0
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1answer
344 views

Time-evolving probability distribution functions with an equation of motion

I came up with this question a while ago and haven't been able to gain any insight on it. You are playing baseball. As a batter with finite vision capabilities, the only information you have about ...
0
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1answer
120 views

A confusing conditional probability problem

$n$ vehicles are stopped at random, the probability that a driver who is stopped is a beginner is $p$ while the probability that a driver who is stopped is a professional is $q$. There are drivers ...
0
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1answer
119 views

What is the joint probability mass function?

A pond has $r$ red fish, $b$ blue fish, and $g$ green fish. Let $R$ be the number of red fish, $B$ be the number of blue fish, and $G$ be the number of green fish in a random sample size of $N$. What ...
4
votes
2answers
358 views

$L^p$ norm of multivariate standard normal random variable

Given $X_i\sim \mathcal{N}(0,1)$ what is the behaviour of $$ ||X||_{l^p}=(\sum_{i=1}^n|X_i|^p )^{1/p}$$ as $n\rightarrow \infty$? For $p=2$ results about $\chi$-distribution tell us that ...
0
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0answers
88 views

Function of random variables

If $X$ is picked from a normalized probability distribution $f(x)$ and $Y$ from $g(x)$, then what is the distribution of $X+Y$ in terms of $f$ and $g$? And that of $XY$? And is there some equation ...
0
votes
1answer
26 views

Taking Limit of a function

I'm reading a proof and I don't understand one step. 1−(1−λ/n)^tn. Now, taking the limit as n→∞, P(Yn/n≤t)⟶1−e^(−λt). How does that work out? In this case, t is ...
2
votes
1answer
145 views

Probability Hyper Geometric Distribution

Can Somone help explain this to me, don't need the answers just a bit of guidance, I'm kind of lost on this one. A consumer advocate claims that 80 percent of cable television subscribers are not ...
0
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1answer
1k views

Cumulative distribution function determine the random variable

I don't know that determine is the right word, but I try to explain. What I need to understand. :) So.. We know's that if a function fit this conditions: Monotonically non-decreasing for each of its ...
0
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1answer
2k views

Distribution of minimum and sum of two independent exponential random variables

How can I solve this problem? Is there any formula for this problem Find the distribution of the random variable $Y$ if $Y=\min(X_1,X_2)$ $Y=X_1+X_2$ where $X_1$ and $X_2$ are independent ...
0
votes
1answer
56 views

Change of Variables and independent random variables.

Suppose that we have two IID random variables, $X_1, X_2$, carried by a triple $(\Omega,\mathcal{F},P)$. While solving an exercise I ended to a point that I had to see that, $$ \iint\limits_D x_1 ...
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0answers
66 views

Need help with calculating covariances of a random process.

Suppose ${X_n; n \in Z}$ and ${Z_n; n \in Z}$ are mutually independent, i.i.d. Gaussian random process with auto correlations $R_x(k) = {\sigma_x}^2\delta(k)$ and $R_z(k) = {\sigma_z}^2\delta(k)$ ...
0
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0answers
61 views

How can I find the distribution of $R$?

We are told that if $X \sim N(0,1)$, then $X^2$ has gamma distribution. Also, if $Y \sim N(0,1)$ and is independent from $X$, then $X^2 + Y^2$ has gammma/chi-squared distribution. Let $(X,Y)$ be a ...
1
vote
1answer
745 views

What is the autocorrelation of a squared Gaussian process?

Suppose $ {X_t; t \in R} $ is a wss, zero mean Gaussian random process with autocorrelation function $ R_X( \tau) ; \tau \in R$ and power spectral density $S_X(\omega); \omega \in R$. If w define ...
4
votes
2answers
155 views

Evolution by death and immigration of Poisson distributed population

This is quite an interesting problem, but I'm not sure how to go about doing it. I know that by using some basic Poisson properties I can figure it out but I'm failing to see how. It goes like this: ...
0
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0answers
50 views

Continuous random varible

If a continuous random variable $X$ is normally distributed with a mean of $11$ and a standard deviation of $0.095$, how do we find the probability that $X > 10.9$? Please help.
0
votes
1answer
228 views

Conditional Expectation Problem with Multiple Variables

Let X be some random variable with unconditional mean 0. Suppose I have that $E[XY]=E[X|Y]=0$ and $E[XZ] \neq 0$ Does it follow that $E[XYZ]=0$? Here are my thoughts: ...
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4answers
3k views

Expected Value of a Binomial distribution?

If $\mathrm P(X=k)=\binom nkp^k(1-p)^{n-k}$ for a binomial distribution, then from the definition of the expected value $$\mathrm E(X) = \sum^n_{k=0}k\mathrm P(X=k)=\sum^n_{k=0}k\binom ...
0
votes
1answer
46 views

simplifying an expression involving an integral

Simplify the following expression $$ \iint_{-\infty}^{c+x}xf(x)f(y) \,dy\,dx+\iint_{c+x}^{\infty}yf(x)f(y) \,dy\,dx $$ where $x$ and $y$ are iid random variables; $c$ is a constant; and $f$ is the ...
0
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1answer
83 views

Interpreting an integral/ probability

Think of two iid random variables $x$ and $y$ with density $f$ and CDF $F$ and a constant $c$. What could the qualitative meaning of the following expression be? ...
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1answer
131 views

What does it mean for a data set to have Gaussian-distributed noise?

I need to find an answer for this question. What does it mean for a data set to have Gaussian-distributed noise? Can anyone help?
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2answers
1k views

Distribution of the sum of a multinomial distribution

I have distilled an error analysis problem into the following: I have a multinomial distribution, $X$, consisting of $n$ independent trials where each trial takes on the values $\{0,1,\ldots,k-1\}$ ...
1
vote
1answer
159 views

Probability of getting any number if I roll the die 4 times.

We have a question to investigate any game between two players that have dice, when the dice are rolled $4$ times what is the probability of getting any number say $4$ or $5$.. note that the highest ...
0
votes
1answer
368 views

Random Variables, Probability question

The question is: A type of algae is distributed in a liquid by the PPP (Poisson Point Process). We know that the number of algae is 2 per liter. Samples of these liquids are provided in containers ...
2
votes
0answers
102 views

Hypergeometric distribution with a random number of draws?

I was wondering if any classical distributions can be arrived at via a hypergeometric r.v. where the number of draws from the urn is also a random variable? Ideally I would like the sample space for ...
1
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2answers
131 views

Probability of first player winning

Two players take turns to toss a coin; the winner is the first to toss a head. What is the probability that the first player to toss the coin wins?
0
votes
1answer
144 views

Find the distribution function of X.

Let the point (u, v) be chosen uniformly from the square 0<=u<=1, 0<=v<=1. Let X be the random variable that assigns to the point (u, v) the number u+v. Find the distribution function of ...
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2answers
112 views

How to describe following set $\{ 1 \leq n \leq N: \alpha_n \in]a,b[\}$?

How to describe the set $$\{ 1 \leq n \leq N: \alpha_n \in(a,b)\}$$ when $(a,b) \subset [0,1)$ and you have following information: a sequence of numbers $(\alpha_1,\alpha_2, \alpha_3,...)$, where ...
0
votes
1answer
98 views

Does only the Gaussian density provide increasing likelihood ratio?

This question is based on the question which I asked before: For which pair of probability density functions, $L=f_1(x)/f_0(x)$ is increasing? however I didnt get any answer other than the Gaussian ...
2
votes
2answers
141 views

Puzzle on Ranks

Here is a puzzle from textbook : 40 Puzzles and Problems in Probability and Mathematical Statistics Peter draws n = 100 independent realizations of a continuous rv and ranks them in increasing order ...