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

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2
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1answer
64 views

Derivation of Poisson from Binomial

I am not very well versed in statistics so any clarification would be appreciated. I understand the mathematical derivation of Poisson from Binomial. I can see just from plotting various Binomial ...
1
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0answers
36 views

Distribution of partial sums of a $L^2$-transformed Gaussian Process

Our assumptions are: $X_t$ is a stationary sequence of standard normal random variables such that $\gamma _X (k)\sim L_{\gamma}(k)k^{2d-1}$ with $d \in (0,1/2)$, where $L_\gamma (k)$ is a slowly ...
0
votes
0answers
164 views

Central Limit Theorem for Dependent Non-Identical Random Variables.

If $X_{(1)}, X_{(2)},\ldots$ are mutually dependent as in the case of ordered statistics and we need to find the sum $S_N$ of all $X_{(i)}$ like $\sum_{i=1}^{N\to \infty} X_{(i)}$. How do we apply ...
0
votes
1answer
160 views

Proof that a sequence of random variables have finite expectation

Let $X_n$ be iid non-negatives random variables. Prove that $\mathbb{E}[X_1] < \infty$ iff $P(X_n \ge n\text{ i.o.}) = 0$ I thought I would start like this for one direction $\infty > ...
1
vote
1answer
54 views

Difference of Poisson r.v's. Is there a simpler way?

Here's a problem I'm having trouble with (section on Poisson r.v.'s): Suppose that when a baby is born, the probability it's a boy is $0,52$ and the probability that it's a girl is $0,48$. On some ...
3
votes
1answer
2k views

Sum of Bernoulli random variables with different success probabilities

Let $X_{i} \in \{0,1\}$ be Bernouli random variable with probability of success $p_{i}$, i.e., $P(X_{i}=1) = p_{i}$ and $P(X_{i}=0) = 1-p_{i}$ and let $Y=\sum_{i=1}^{n}X_{i}$ for $n>0$. Is it ...
4
votes
0answers
365 views

Simulating from a Multivariate Gaussian without Cholesky

I'd like to draw a sample from a multivariate Gaussian distribution $\mathcal{N}(\mu, \Sigma)$, where $\mu$ is the mean vector (can assume it to be $\boldsymbol{0}$), and $\Sigma$ is a sparse positive ...
2
votes
0answers
37 views

Inferring a probability distribution from another probability distribution

Let $A$ and $B$ be real-valued random variables, with $f_A$ and $f_B$ their probability density functions. Let's say we can observe the values of $A$ many times and estimate $f_A$ fairly precisely. We ...
1
vote
3answers
46 views

Doubt about why I can't treat this as a Bernoulli process

I know the title is not descriptive enough, but I don't know how else to say it. I don't know why I can't use the Binomial distribution to get the result I'm looking for. The teacher solved it long ...
2
votes
3answers
449 views

prove that any positive integer-valued random variable with memoryless property has the geometric distribution for some $p$

How to prove that any positive integer-valued random variable with memoryless property has the geometric distribution for some $p$. By memoryless property, $$P(X=i+s | X>i)=P(X=s)$$ How to get ...
1
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0answers
40 views

limit distribution of possion distribution

Assume Xn is possion with mean $\lambda_n$ and suppose that $\lambda_n\rightarrow\infty$ as $n\rightarrow \infty$. Then how to show Xn is AN$(\lambda_n,\lambda_n)$. I've tried to use characteristic ...
0
votes
1answer
155 views

Integrate over the uniform distribution on the simplex

Let $p=(p_1,\ldots,p_n)$ correspond to points in a simplex that add up to one, i.e. $p$ is a discrete probability distribution. I would like to compute an integral of the form $\int dp_1\ldots\int ...
0
votes
4answers
53 views

A basic doubt on the definition of a Poisson random variable

What is the significance of "large city" in the definition of the following Poisson variable : "Number of phone calls placed during a ten second interval in a large city" I guess either $n \to ...
2
votes
0answers
265 views

Probability distribution of the product of two independent complex gaussian random variables

I have to calculate the pdf of $Z = X*Y$, where $X \in \mathcal{C}(\mu_x,\Sigma_x)$ and $Y \in \mathcal{C}(\mu_y,\Sigma_y)$, where $\mathcal{C}$ is a complex distribution. It can be assumed that ...
1
vote
1answer
71 views

finding the probability density function of $ dY_t = - Y_t X_t dW_t$

Could someone point me to where I can learn how to derive the stationary distribution for the martingale $Y_t$ which itself has stochastic volatility drive by $X_t$: \begin{align} dY_t &= - Y_t\ ...
0
votes
1answer
78 views

Poisson process of alternating sources, pairing products of the two of them

I have a problem I've been thinking for a while, but it's confusing me quite a lot. There are two independen machines, A and B. A produces a product at a rate of 2/minute, and B does it at a rate of ...
4
votes
2answers
193 views

A Bernoulli trials problem

Two parents decide to have children until they have 3 children of the same gender one after another (3 in a row). If p(boy)=p(girl)=1/2, how many children are they expected to have? I have tried to ...
0
votes
2answers
901 views

weighted sum of exponential random variables

Suppose $X_i$ are i.i.d random variables and $X\sim\operatorname{Exp}(\lambda)$ i.e. $Pr(X\le x)=1-e^{-\lambda x}$ for $x \ge 0$. What is the density function of $Z=\sum_{i=1}^{N}\alpha_iX_i$ where ...
1
vote
0answers
49 views

Almost Surely Convergence

I need some help with computing the lim inf and lim sup of $ \frac 1n \sum_i X_n$ where the density of variable $X_n$ is absolute continuous, say, f(x) = exp(-x). I am interested in using the ...
3
votes
3answers
175 views

$E[X]$ finite iff $\sum\limits_{n} P(X>an)$ converges

Show that: $$\sum\limits_{n \in N } P(X>an) < \infty\ \text{for some}\ a > 0 \Rightarrow E[X] < \infty \Rightarrow \sum\limits_{n \in N } P(X>an) < \infty\ \text{for every}\ a > ...
1
vote
2answers
45 views

$P\left( n,\left( {{\lambda }_{1}}+{{\lambda }_{2}} \right)T \right)$ Disaggregating Tail of Poisson

I have a Poisson tail $P\left( x,\left( {{\lambda }_{1}}+{{\lambda }_{2}} \right)T \right)$ which is sum of two independent Poisson distribution with rate $\lambda_1$ and $\lambda_2$. I am trying to ...
0
votes
2answers
475 views

If $p_1 = 0.3$ and $p_2 = 0.4$, what is the probability that it will take Jay more than 12 hours to be successful on both jobs?

Jay has two jobs to do, one after the other. Each attempt at job $i$ takes one hour and is successful with probability $p_i$. If $p_1 = 0.3$ and $p_2 = 0.4$, what is the probability that it will take ...
1
vote
1answer
34 views

Estimating a probability with converging moments

Let me rephrase my question. If you look at the random variable $X$ which simply picks a random integer between $1$ and $N$ (distributed uniformly) and now look at the inequality $$ t^k \cdot ...
1
vote
1answer
112 views

The weighted distribution function for combination of two variables

For example, we have two random variables $a$ and $b$. And they have cumulative distribution function $F(x)$ and $H(x)$. We have number $0 < p < 1$. Suppose, some machine get this random ...
1
vote
1answer
135 views

what will be the procedure to prove the following relationship?

Let $U$ follows standard uniform distribution , that is, $U\sim U(0,1)$ and $X$ follows Pareto distribution, that is, $X\sim Pa{(\alpha,a,h)}$ where , $a=$location parameter ; $-∞<a<∞$ ...
3
votes
3answers
92 views

Probability density function of $W = X + Y$ where $X \sim \mathrm{Unif}[0,1]$ and $Y \sim \mathrm{Exp}(\lambda)$ are independent random variables?

This is the question that I was given: And this was the provided solution: I can't seem to make sense of it - firstly, what exactly is the question asking you to do? How did they know to divide ...
2
votes
0answers
27 views

Multivariate Distribution Question?

If $(X,Y)$ have the following joint distribution: $$f_{X,Y}(x,y) = \begin{cases} 2 f_X(x)f_Y(y) & \text{if }xy>0 \\[6pt] 0 & \text{otherwise} \end{cases} $$ where $f_X(·)$ and $f_Y(·)$ ...
1
vote
0answers
80 views

P.d.f of a discrete fourier transform of binary variables

Let $\{a_n\}$ be a set of $N$ "binary" random variables uniformly distribuited in $\{-1,1\}$. The discrete fourier transform is defined $b_k=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1} a_n e^{-2 \pi i k n ...
3
votes
2answers
112 views

probable squares in a square cake

There is a probability density function defined on the square [0,1]x[0,1]. The pdf is finite, i.e., the cumulative density is positive only for pieces with positive area. Now Alice and Bob play a ...
0
votes
4answers
75 views

Probability of $X$ out of $N$ dice landing on $M$

The problem is as follows: We have $N$ dice and we throw them on a table. What is the probability that $M$ will fall $X$ times? Specific example: We have $10$ dice and we throw them on a table. What ...
0
votes
1answer
27 views

Length of life of a fire detector

The length of life of a flame detector is exponentially distributed with paramater $\lambda=0.1/year$. Die number of events which activate the flame detector in an interval with length $t$ (heat, ...
6
votes
2answers
881 views

The probability of a drunk person/random walk

A drunk person wonders aimlessly along a path by going forward 1 step and backward 1 step with equal probabilities of $\frac12$. a) After 10 steps, what is the probability that he has moved 2 steps ...
0
votes
1answer
210 views

Finding probability that a person gets $7$ when rolling a pair of dice

*I STILL DON'T GET THE ANSWERS PROVIDED. PLEASE HELP! In a game, the participant rolls a pair of dice. If the result is a $7$, he wins. If the outcome is a number $n$ different from $7$, he continues ...
1
vote
3answers
108 views

How I can find the expected value of $G$?

Suppose two teams play a series of games, each producing a winner and a loser, until one team has won two more games than the other. Let $G$ be the total number of games played. Assuming each ...
0
votes
1answer
17 views

normal vector condition on 1 component

If you have two normal random variables $Z_1$ and $Z_2$ possibly correlated namely you have a multivariate normal distribution $Z= (Z_1, Z_2)$ What is the conditional distribution $(Z_1|Z_2 =a)$? ...
0
votes
0answers
77 views

Spatial distribution of bees

* Please please help! I still get stuck. We have a forest for bees, consisting of $4$ non-overlapping regions. $80\%$ of the bees seek honey in the forest while $20\%$ of the bees do so outside the ...
0
votes
1answer
44 views

find the distribution

Suppose two teams play a series of games, each producing a winner and a loser, until one team has won two more games than the other. Let G be the total number of games played. Assume each team has a ...
0
votes
0answers
36 views

identifying the distribution and finding expectation

Suppose there are n boxes labelled $1, 2,\ldots, n$ and $n$ balls labelled $1, 2,\ldots,n$. Balls are placed at random in the boxes. Let $X$ be the number of empty boxes. Find $E(X)$ and ...
1
vote
1answer
67 views

Moment generating function of two non-independent Brownian increments

I am writing to ask if it is possible to get closed-form solution to the expression to the following expression: $\mathbb{E}[e^{\sigma^2(W_t-W_u)(W_s-W_u)}]$ where $W$ is a standard Brownian motion, ...
0
votes
3answers
370 views

SOA/CAS Exam P Question (from previous exam (Nov. '09)): Finding percentiles

So this question comes from SOA/CAS Exam P of November 2009, I'm not sure why the solution is the way it is. The question says this: An insurance company sells an auto insurance policy that ...
1
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0answers
30 views

Truchet tiles on a cube [duplicate]

We randomly place copies of the tiles into faces of the flattened cube. 1.Find the probability that the circular arcs on the Truchet tiles will form one loop, two loops, three loops and four loops? ...
1
vote
2answers
91 views

Summation of independent discrete random variables?

We have a summation of independent discrete random variables (rvs) $Y = X_1 + X_2 + \ldots + X_n$. Assume the rvs can take non-negative real values. How can we find the probability mass function of ...
5
votes
2answers
96 views

finding the number of circles we get when randomly placing given patterns into a grid of squares

We have an 11$\times$11 table of squares (consist of 121 squares of dimension 1$\times$1). we have 3 tiles shown in the picture. Each tile has dimension 1$\times$1. we now randomly pick 3 tiles into ...
1
vote
1answer
337 views

Trouble understanding sum and product of probability distributions

Having trouble understanding where can we use the sum and product of probability distributions. Could someone please provide me with a real-life scenario? I think this is what I need to understand the ...
1
vote
1answer
38 views

Question on Poisson Processes

Good evening, I am sort of stuck in one problem of Poisson Processes and I hope I could get some help (no it is not a homework). Suppose that the customers arrive at the ticket booth independently. ...
6
votes
1answer
430 views

How was the normal distribution derived?

Abraham de Moivre, when he came up with this formula, had to assure that the points of inflection were exactly one standard deviation away from the center, and so that it was bell-shaped, as well as ...
3
votes
1answer
1k views

How do I sum two Poisson processes?

If we have a Poisson Process $Y$ with intensity $\lambda$ and a Poisson Process $X$ with intensity $\mu$, where $X$ and $Y$ are two independent Poisson processes. How can I find the process ...
1
vote
1answer
74 views

A modified Buffon's needle

A needle 2.5cm long is dropped on a piece of paper that has a very fine parallel lines 2.25cm apart drawn on it. What is the probability that the needle lies between the two lines? I can see how ...
-4
votes
1answer
62 views

this is about prabability [duplicate]

Suppose a multiple choice test consists of $100$ questions and each question has $5$ possible answers, only one of which is correct. Four points are awarded for each correct answer, and $1$ point is ...
2
votes
2answers
158 views

Other way to express $e^{|x|+|y|}$

I have a joint PDF with $e^{|x|+|y|}$. I know I can separate the function in two functions, $e^{|x|}$ and $e^{|y|}$. The limits for $x$ and $y$ are from $-\infty$ to $\infty$. Can I integrate from $0$ ...