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

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3
votes
1answer
48 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 ...
0
votes
0answers
8 views

Is it possible to get the PDFs of each of the three vector components knowing the PDF of the modulus if isotropy is guaranteed?

The PDF of a given vectorial quantity modulus is known. I would like to obtain the PDF of each of the three vector components in the case of isotropy, i.e. the three PDFs are supposed to be equal and ...
0
votes
0answers
35 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 ...
2
votes
1answer
29 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
votes
1answer
37 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
votes
1answer
14 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
vote
3answers
59 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
25 views

probability-distribution that has its mode equal median

Could anyone tell me any asymmetric distribution whose mode=median? Thanks in advance.
1
vote
2answers
32 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
vote
1answer
42 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
vote
0answers
32 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
votes
1answer
44 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
votes
2answers
130 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
vote
2answers
37 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
votes
0answers
15 views

Vysochanskij Petunin vs. Cantelli inequality for random variables

The well known Cantelli inequality states: $$Pr(|X-\mu|\ge\alpha)\le\frac{2\sigma^2}{\sigma^2+\alpha^2}$$ where $X$ is a real valued random variable, $\mu$ the mean value and $\sigma^2$ the variance ...
0
votes
1answer
23 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
vote
1answer
19 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
votes
1answer
38 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
vote
0answers
35 views

Sums of Power Law random variables

Suppose $F$ be a pareto distribution with scale parameter $x_m$ and shape parameter $\alpha$. Assume $X_1, X_2 , ..., X_n$ are iid random variables drawn from $F$. Let $S_n(k) = X_1 ^k + X_2 ^k + ...
0
votes
1answer
48 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 ...
2
votes
3answers
39 views

Expected value of moment generating functions: [closed]

How do I do these? I don't understand any of them.. especially the last two. I'm studying for a final soon and need help. I recognize that they are distributions but how do I answer the question?
3
votes
1answer
57 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 ...
1
vote
0answers
37 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
vote
1answer
29 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 ...
0
votes
1answer
15 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 $\sqrt{ m/n }$, ...
0
votes
1answer
32 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
votes
1answer
24 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 ...
1
vote
0answers
24 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
vote
0answers
48 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*} ...
0
votes
1answer
35 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
vote
1answer
29 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, ...
1
vote
0answers
50 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 ...
0
votes
0answers
18 views

The distribution of ratio of two shifted gamma

I am wondering if anyone can help me to find the ratio of this distribution. Assume $S$ and $T$ are independent, where $S\sim Gamma(n-1/2, 4(1+\rho)\sigma^2)$ $S\sim Gamma(n-1/2, ...
1
vote
1answer
36 views

Square root of Chi-square distribution tends to $N(0,1)$

The question requires to show that $\sqrt{2\chi^2_n}-\sqrt{2n}$ converges in distribution to $N(0,1)$ as $n \rightarrow \infty$, which I dont know how to proceed. The question also has a first part ...
0
votes
1answer
32 views

Expected value of series of uniformly converges random variables [duplicate]

Let $X_1,X_2,X_3,...$ a series of i.i.d. variables with $X_i \sim \mathcal{U}(0,1)$. Let $N=\inf\{n\mid \sum_{i=1}^{n}X_i\geq1\}$ Prove that $E(N)=e$. I don't really have a clue how to even start ...
0
votes
0answers
29 views

Finding $p$ of the binomial cdf…

Please bear with me, I'm only a biologist ^.^: I have a need of solving this cdf so as I can plug in known values $Pr, n, k$, and get an answer for $p$. $$f(k;n,p) = Pr(X\le k) = \sum_{i = ...
2
votes
1answer
35 views

Why is this distribution Poissonian?

Do this experiment. Draw 10000 random number in $[0,1]$ according to the uniform distribution. Order them in the increasing order. The difference between two neighbouring numbers follows a Poisson ...
0
votes
1answer
34 views

What is the distribution of the product between a continuous (Exponential) and a discrete (Bernoulli) random variables?

If you have $X$ distributed as a $\mathrm{Bernoulli}(p)$ and $Y$ as a $\mathrm{Exponential}(\lambda)$ find $Z=XY$. I tried doing it with the MFG i.e. ...
0
votes
0answers
51 views

How to get the Probability

We have two corresponding ropes, with 50 length units for each one, and there's a labeled homologous area on the ropes.. with 17 length units for each area.... Now we are going to perform the ...
1
vote
1answer
30 views

Probability - A trial consists of tossing a fair coin twice and noting H = number of heads observed…

A trial consists of tossing a fair coin twice and noting H = number of heads observed. What is the probability that if 5 trials are performed, we will note H=0 two times, H=1 one time, and H=2 two ...
0
votes
0answers
37 views

How to show $ P\big(\big|\frac{X}{n}-p\big|>a\big)\le\frac{\sqrt{p(1-p)}}{a^2n}min\big\{\sqrt{p(1-p)},a\sqrt{n}\big\}$

Let $X$ be binomial, $B(p,n)$ with $p>0$ fixed, and $a>0$. Show that, $\displaystyle ...
1
vote
0answers
11 views

Counterexample: Convergence in finite dimensional distributions does not imply weak convergence

I'm working at the following exercise: Give an example of a sequence of stochastic processes $(\mathbb{X}^n)_{n\geq 1}$ such that the finite dimensional distributions converge to the finite ...
-2
votes
1answer
32 views

what is the probability that it was lost by second post-office? [closed]

parcels from sender S to receiver R pass sequentially through two post-offices. Each post-office has a probability of 1/5 of losing an incoming parcel, independently of all other parcels.Given that a ...
0
votes
0answers
35 views

Approximate the distribution of the sum of ind. Beta r.v

If $X_i$ has a Beta distribution $\beta(1,K)$. What is the best approximation for the distribution of $ S=\sum_{i=1}^N X_i$, when the $X_{i}$ are independent and $N$ is finite. Thanks
-3
votes
1answer
56 views

Find the expectation $E[X]$ [closed]

Let $X$ be a random variable which is uniformly chosen from the set of positive odd numbers less then 100. Find the expectation $E[X]$?
0
votes
0answers
31 views

The amount of time that a customer spends waiting at an airport check-in counter is

The amount of time that a customer spends waiting at an airport check-in counter is a random variable with mean 8.1 minutes and standard deviation 1.5 minutes. Suppose that a random sample of n=46 ...
1
vote
1answer
54 views

Prove that $Z =\sum_{j=1}^{n}S_{j}$ is distributed as $\mathrm{NB}(\frac{n\alpha}{\ln(1-\delta)},1-\delta)$

Prove that $Z=\sum_{j=1}^{n}S_{j}$ is distributed as $$\mathrm{NegativeBinomial}\left(\frac{n\alpha}{\ln(1-\delta)},1-\delta\right)$$ Being $S_{j}=\sum_{i=1}^{N}X_{i}$ where $N$ is distributed as ...
1
vote
1answer
32 views

Intuition wrong? Variance of two related sums of random variables.

I recently have been trying to analyze the variance of two related random variables $G$ and $F$, hoping to get a decrease of the variance when switching to the second ($F$). However, either my ...
0
votes
0answers
8 views

Limiting expression for Power law tail index from a quantile function?

Assume that I have a random variable $X$ (which I know will have a power law tail). If I had the CDF for $X$, $G(x)$, then I could easy calculate this tail as something like, $$ \alpha = ...
0
votes
1answer
18 views

What is probability-frequency function?

What is probability-frequency function and how it differs from probability function?