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

learn more… | top users | synonyms

1
vote
0answers
5 views

Laplace transform of stopping times

I am nearly done with a question: Let $(B_t)$ be a Brownian motion on $\mathbb{R}$. For a fixed $x >0$, let $\tau$ be a stopping time defined by $$ \tau = \inf \{t \geq 0 : B_t \not \in (-x,x) ...
2
votes
2answers
18 views

Pdf of an inverse of random variable

Assume that we have a pdf f(x) with random variable X. How can you find the pdf of the random variable Y = 1/X ? I checked some examples including Cauchy distribution and I saw Jacobian in Cauchy ...
0
votes
1answer
25 views

$P(\min(X_1,\dots,X_n) > t) = P(X_1>t,\dots, X_n>t)$

One step in the my solutions book shows... $P(\min(X_1,\ldots,X_n) > t) = P(X_1>t, \ldots, X_n>t)$, where $X_1, \ldots, X_n $ are independent and $X_j \sim \mathrm{Expo}(\lambda) $ Why is ...
0
votes
1answer
26 views

Deducing means of normal distributions

I have a set of objects ($i$) with different lengths ($x_i$), and two devices ($a$ and $b$) to measure them. Each device measures with an error which probability density function (pdf) follows a ...
0
votes
1answer
117 views

Independent, Uniform, Random Variable:

Working on this: Alice and Bob agree to meet in the Copper Kettle after their Saturday lectures. They arrive at times that are independent and uniformly distributed between 12:00 and 13:00. ...
0
votes
0answers
17 views

Integral of a Poisson-Exponential Joint Distribution

I'm considering a joint distribution of Exponential variable $X$ and Poisson variable $Y$. the thing is, I can't figure out a way to derive the $pdf$ for such a distribution. I know that ...
3
votes
2answers
48 views

Convergence to $N(0,1)$ in distribution

Question is as following: $X\sim Po(\lambda)$ $$\frac{X-\lambda}{\sqrt{\lambda}} \,{\buildrel d \over \rightarrow}\, N(0,1)$$ as $\lambda \rightarrow \infty$. Obs. One is asked not ...
0
votes
1answer
19 views

Data distribution for sine time series

Suppose we have a time series $x_t=\sin(0.02\pi t)$. Although this time series is totally deterministic, we can treat it as one realization of a proto/quasi/pseudo-stochastic process and estimate the ...
-2
votes
1answer
14 views

Distribution of a Gaussian Random variable vector [on hold]

I read a slide on the internet which show that: If the random vector $w\sim N(0,I )$ then how can I prove: $x= A^{1/2}w+\bar{x}$ has the distribution $N(\bar{x},A)$ Here A is the covariance matrix ...
1
vote
1answer
396 views

Solving Probability Density Function for continuous random variable

The probability density of a random variable $x$ is $$f(x)=a\ \cdotp x^2\ \cdotp \mathrm{e}^{−kx}\ (k>0,\ 0\leq x\leq \infty)$$ Then, the coefficient $a$ equals $$(i)\frac{k^3}{2}\ \ \ \ (ii)\ k^3 ...
0
votes
0answers
16 views

Suppose X follows a binomial distribution with parameters n=100 and p=1/5,then prove that P(X=r) is maximum when r=33 [on hold]

smeone plz solve this question Q.Suppose X follows a binomial distribution with parameters n=100 and p=1/5,then prove that P(X=r) is maximum when r=33. aren't we required to find the mode here which ...
1
vote
2answers
44 views

Let $X$ be Hypergeometric, Find $E\left(\binom{X}{2}\right)$

Let X be Hypergeometric: $X \sim \operatorname{HGeom}(w,b,n)$, so that $X$ is the number of white balls in a sample of size $n$ out of a population of $w+b$ white and black balls. Find ...
6
votes
1answer
140 views
+100

scores of individuals and evaluation

Suppose we have a fixed (ordered) set of $2000$ integers $p_m$ drawn from a discrete uniform distribution on $\{1,2,...,100\}$ arranged in a terrain. Let this terrain be denoted $\mathcal{T} = ...
1
vote
0answers
11 views

Error analysis of kernel density estimation

Let $X$ be a random variable with true density $f$, $Y = \{y_i\}_{i=1}^n$ be a realization of a random variable in $d$-dimensional space $R^d$, and $\hat{f}$ is the density estimator of $Y$ using ...
1
vote
1answer
28 views

Position of Brownian motion at exit time from the upper half plane

I am currently reading some books on SLE and struggling on some problems regarding Brownian motion. For a Brownian motion in $\mathbb{R}^2$ starting from $(x,y)$, I don't know how to find the ...
3
votes
1answer
24 views

Hitting time process of Brownian motion [on hold]

I am stuck with this problem: Let $(B_t)$ be a standard Brownian motion in $\mathbb{R}$. For $t \geq 0$, let $$ H_t = \inf \{ s \geq 0 : B_s = t \}, \quad S_t = \inf \{ s \geq 0 : B_s > t \}. $$ ...
0
votes
1answer
369 views

How do I find if the probability of the sample proportion is greater than something?

I have this problem and I have no clue how to solve it. In 2012, 31% of the adult population in the US had earned a bachelor’s degree or higher. One hundred people are randomly sampled from the ...
-1
votes
0answers
21 views

Distrubution of the maximum of a sequence of random variables. [on hold]

Let $\{Xn\}_n$ a sequence of independent random variables with the same distribution. Suppose their mean is 0 and their variance is 1. Consider $$\max_{0\le k \le n} X_k(t)-tX_k(1)$$ How can I ...
0
votes
0answers
16 views

Question about upcrossings of Brownian motion

I am very stuck on this problem: Given a Brownian motion $(B_s)$, write $S_t = \sup_{0 \leq s \leq t} B_s$, for each $t>0$. All stopping times and martingales are considered w.r.t the filtration ...
2
votes
1answer
40 views

Conditional expectation of an uniformly distributed random variable

Suppose $U_1, \ldots, U_n$ are i.i.d. random variables with $U_1$ distributed uniformly on the interval $(-1, 1)$. Compute $\mathbb{E}(U_1 + \ldots + U_n |\max(U_1, \ldots, U_n) = t)$ for $t \in (-1, ...
0
votes
0answers
30 views

Three state probability where one state has yet to factor results. [on hold]

I'm currently trying to explain something to someone else using probability to make it simpler to understand. As I have it now there have been 5 examples that have happened. In one case there are 3 ...
-1
votes
1answer
46 views
+100

Probability that there is sub-sequence of exact length

Can you help me to solve the following: Find probability that in sequence of N random uniformly distributed numbers there is increasing sub-sequence of exact length L.
0
votes
1answer
12 views

definition of Cumulative distribution function

let X be RV, and his Cumulative distribution function: there is a difference if in my case if $X<x$ ? the definition is the same?
2
votes
0answers
46 views

Calculate expected values of the lengths of line segments

There is a line segment of the length of $1$. $N-1$ points are randomly chosen in it, so it is divided by $N$ parts. The question is to calculate expected values of all these parts, from the shortest ...
0
votes
1answer
24 views

Two series of independent Bernoulli trials. Find distributions of being simultaneously successful and of first success being simultaneous.

Nick and Penny are independently performing independent Bernoulli trials. For concreteness, assume that Nick is flipping a nickel with probability p1 of Heads and Penny is flipping a penny with ...
1
vote
2answers
22 views

probability of a brownian motion being equal to the running maximum

Let $B$ be a standard Brownian motion on $\mathbb{R}$. I would like to show that $$ \mathbb{P} \bigg\{ B_1 = \max_{t \in [0,1]} B_t \bigg\} =0 .$$ I argue that since $\max_{t \in [0,1]} B_t $ has the ...
0
votes
1answer
21 views

How is logistic loss and cross-entropy related?

I found that Kullback-Leibler loss, log-loss or cross-entropy is the same loss function. Is the logistic-loss function used in logistic regression equivalent to the cross-entropy function? If yes, can ...
0
votes
1answer
36 views

Joint distribution of arrival times in Poisson process

I need to compute the following joint distribution in a Poisson process: $f_{S_A S_{A+B}}(t_1, t_2), t_2\ge t_1$ $S_A$ and $S_{A+B}$ are the arrival epochs of the $A^{th}$ and ${A+B}^{th}$ arrivals ...
1
vote
1answer
106 views

Limit of Beta distribution on $[0, A]$ as $A\rightarrow \infty$ with constant expectation and variance

I am trying to determine the limiting form of a beta distribution as its range expands under isoparametric constraints on its first two moments.... For reference $X_A \sim Beta(0,A,\alpha,\beta) = ...
0
votes
2answers
31 views

Conditional distribution of geometric variables

Setting Suppose X1 and X2 are independent with the common geometric distribution w(k; p). Determine the conditional distribution of X1 given that X1 + X2 = n. Solution My argument is $$\Pr[X_1| ...
1
vote
1answer
30 views

Who can help me to interpret the following expression?

I don't understand the second line of the following expression. Why does he use conditional expectation? and can you explain the following calculation process to me? Thanks.
1
vote
0answers
21 views

How to calculate the following conditional expectation? Is my calculation process right?

I want to calculate the conditional person's correlation coefficient. But I don't know how to calculate the following expressions,especially the conditional expectation of ...
1
vote
2answers
40 views

Selecting n matches from two pockets.

Setting An eminent mathematician fuels a smoking habit by keeping matches in both trouser pockets. When impelled by need he reaches a hand into a randomly selected pocket and grubs about for a match. ...
0
votes
2answers
35 views

Determine the probability distribution of a ratio of two random variables?

Setting You are given two independent random variables $X_0,X_1$ with common exponential density $f(x) = \alpha e^{-\alpha x}$. Let $R = \frac{X_o}{X_1}$. Determine $\Pr[R > t]$ for $t > 0$. I ...
0
votes
1answer
27 views

Using Poisson distribution to find the probability that the interval between arrivals exceeds some value

Suppose we have a helpdesk with tickets arriving at a rate of three per min. Tickets arrival follow a Poisson distribution. How someone can calculate: a. The probability of the time between the first ...
0
votes
0answers
11 views

Limit distribution on return time $\tau = \inf\{k: X_k = X_m \text{ where }m<k\}$ [on hold]

Suppose there is a stochastic process ${X_i}_{i=1}^n$ where $X_i$ is distributed normally over $\{1,\dots,n\}$. As $n\rightarrow \infty$, the probability that any one value is repeated should go to ...
0
votes
0answers
10 views

Iso-density locus of Gaussian mixture distribution

I would like to known what is the equation of the iso-density locus of a Gaussian mixture distribution. Is such an iso-density locus a union of ellispoids? Let's say that this Gaussian mixture is in ...
5
votes
1answer
342 views

Memoryless property and geometric distribution

Suppose $X$ is a random variable taking values in $\mathbb N_0$ with the memoryless property,i.e., for each pair of number $s,t \in \mathbb N$, $$P(X\geq s+t\mid X>t)=P(X\geq s)$$ Show that a ...
0
votes
2answers
402 views

Poisson arrivals during an exponentially distributed interval

This is a marked homework question, so please try not to write complete solutions here: The number of customers that arrive at a service station during a time t is a Poisson random variable with ...
0
votes
1answer
24 views

Determine the density of sum of three normal variables.

Setting $\pmb{X} = (X_1,X_2,X_3)$ is a properly center normal with covariance matrix $$\begin{pmatrix} a & b & 0\\ b & d & 0\\ 0 & 0 & e \end{pmatrix}$$ Determine the ...
0
votes
1answer
25 views

Probability distribution for putting balls in boxes in a correlated way

I'm looking for help finding a probability distribution: Right now I have a problem where I have N indistinguishable balls, which I need to put into K indistinguishable boxes, each of which can hold ...
0
votes
0answers
21 views

AI Bayes Network Question? [duplicate]

A) Given this Bayes Net Answer and explain: 1) True or False 2) True or False B) Given this Bayes Net: Answer and explain: 3) True or False 4) True or False
0
votes
1answer
36 views

Binomial distribution giving me an answer above 1?

I am doing the following question. If i have a box of $20$ soccer balls and the independent chance of a soccer ball of being flat is $0.1$. What is the probability of having at least $4$ flat soccer ...
1
vote
1answer
27 views

distinguishing probability measure, function, distribution

I have a bit trouble distinguishing the following concepts: probability measure probability function (with special cases probability mass function and probability density function) probability ...
3
votes
1answer
23 views

Probability measures and stochastically dependent events

If $P(B\mid A) > P(B)$ and $P(C\mid B) > P(C)$ can I infer that $P(C\mid A) > P(C)$? My suspicion is yes but I don't see how to prove it yet.
0
votes
3answers
26 views

probability of the empty set for arbitrary probability measures

I have a probability space $(\Omega, \mathcal{P}(\Omega), P)$. I want to know the probability of the empty set $\{\}$. Intuitively, I would say this probability is zero. It certainly is for the ...
1
vote
1answer
43 views

Using the geometric distribution to find the probability that between 4 and 6 devices will be tested

Quality control tests spark plugs until they find one that doesn't work. If the probability of a spark plug working is 0.99, what is the probability that they will test between 4 and 6 (inclusive) ...
0
votes
2answers
17 views

Product of two Beta distributed random variables

I have two Beta distributed random variables : $X_1=B(\alpha_1, \beta_1)$ $X_2=B(\alpha_2, \beta_2)$ What can we say about $Y=X_1.X_2$? Is this also a Beta distributed random variable?
4
votes
0answers
37 views

Convergence of $n^{-\gamma}T$ where $T$ a hitting time for uniform rvs, can I use CLT?

Let $X_1,X_2,\dots$ be iid uniform on $\{1,\dots,n\}$ and define $T=\inf\{k:X_k=X_r \text{ for some }r<k\}$. The objective is to figure out when $n^{-\gamma} T$ converges weakly to some ...
0
votes
2answers
31 views

Does this integral $\int f_{X|Y}(x|y) dy$ has any meaning in probability or statistics

Suppose I have two random variables $(X,Y)$ with joint probability density given by $f_{X,Y}(x,y)$. Does integral \begin{align*} \int f_{X|Y}(x|y) dy \end{align*} evaluate to something or has ...