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

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2
votes
3answers
56 views

Distribution of $R^2 = X^2 +Y^2$ where (X,Y) is a point on the unit circle

So I have a point $(X,Y)$ chosen from the unit disk with uniform distribution. And I'm attempting to find the distribution of $R^2$, where $R$ is the distance from the point to the origin. Now ...
2
votes
1answer
25 views

Use cdf to find expectation

I have a cdf for a $\mathbf {discrete}$ random variable, $X$, $$F_X(x)=1-(1-p)^{xn}$$ where $p\in(0,1)$, $n\in\mathbb N$, $x\in\mathbb N$ My thought is to use $$E[X]=\sum_{x=0}^\infty ...
3
votes
3answers
69 views

Suppose that $X$ has the exponential distribution. Find the density for $X^3$

Suppose that $X$ has the exponential distribution. Find the density for $X^3$. Really not sure where to go with this problem, my notes from class weren't sufficient and after poking around online ...
0
votes
1answer
23 views

Conceptual/Notational question on conditional distributions and “given”

So in the book I'm reading, I see the notations $f(x|\theta)$ being used to refer to population distributions, dependent on $\theta$ which are in a family. The author explains this as a notational ...
2
votes
2answers
84 views

Splitting Poisson process formal proof

Let $\{X_t\}_{t\ge 0}$ be a Poisson Process with parameter $\lambda$. Suppose that each event is type 1 with probability $\alpha$ and type 2 with probability $1-\alpha$. Let $\{X^{(1)}_t\}_{t\ge 0}$ ...
3
votes
1answer
32 views

Probability distribution of tossing a coin until obtaining $ k$ heads in a row

In this Question the correct answer is the negative binomial distribution. My problem is: What is the distribution if I want the k heads in a row? Any help is apreciated.
3
votes
2answers
36 views

How to generate a random number from a pareto distribution

I'm working on a problem where I am trying to generate a random number from a Pareto distribution. Using some measured data, I have been able to fit a Pareto distribution to this data set with ...
0
votes
1answer
26 views

Show that $R^{2}$ and $\theta$ are independent and $R^{2}\sim U(0,1)$, $\theta\sim U(0,2\pi)$ in marsaglia's method

I have little problem to show that $R^{2}$ and $\theta$ are independent in marsaglia's method and furthermore $R^{2}\sim U(0,1)$ and $\theta\sim U(0,2\pi)$. For the first method (Box & Muller) ...
0
votes
1answer
39 views

Assume that $X,X_1,X_2,…$ are iid with characteristic function $\phi(t)=\mathbb E[e^{itx}]$, and let $S_n = X_1 + X_2 + X_3 + …$.

Assume that $X,X_1,X_2,...$ are iid with characteristic function $\phi(t)=\mathbb E[e^{itx}]$, and let $S_n = X_1 + X_2 + X_3 + ...$. (a) For a random variable $X$, $X$ and $-X$ have the same ...
2
votes
1answer
29 views

Compute the cumulative distribution function of the variable $R=\sqrt{X^2+Y^2}$

I've returned to the study of statistics after a long while and I'm trying to solve some problems. One of those is the next: Suppose $X$ and $Y$ are random independent variables with normal ...
1
vote
0answers
20 views

Show that an event has strictly positive probability

Consider the random variables $W_i,W_j, X_i, X_j$ with $X_i\sim X_j$, $X_i\perp X_j$ and $W_i\sim W_j, W_i\perp W_j$, where $\sim$ denotes equal probability distribution and $\perp$ denotes ...
1
vote
3answers
45 views

Probability of number of different suits when choosing three cards from a deck

When you pick three cards, without replacement, from a standard 52 card deck, what are the probabilities of: only one suit in your three cards two different suits in your three cards three different ...
0
votes
0answers
11 views

Composition of Binomial Distributions/Normal Approximation

I'm modeling as system as follows: $X_{0} = 1$ $X_{t+1} = X_{t} + Z_{t}$ $Z_{t} \sim Binomial(n=X_{t}, p)$ i.e. a composition of binomial distributions I'm interested in the variance of $X_{t}$ ...
1
vote
2answers
28 views

Which definition is correct for a geometric random variable?

Is it The number of failures BEFORE the first success OR The number of trials required to get a first success? Also, if I was to work out the expected value of a geometric random variable, say $p ...
0
votes
0answers
13 views

Doubts in Ignatov's Theorem proof (Sheldon M. Ross Book)

Good morning, I am working in Ignatov's theorem and I have a doubt regarding the proof which can be seen in Sheldon M. Ross book (Introduction to probability models). We have a sequence of ...
1
vote
3answers
44 views

How does using one distribution as another's sample size affect variance?

How does using one distribution as another's sample size affect variance? For example, let's say I roll a 6-sided dice and record the number shown. Then, I roll 'that many' 6 sided dice more and ...
0
votes
1answer
26 views

Generating correlated standard normals

Suppose I want to generate three standard normals $X, Y, Z$ with correlation matrix given by $R$= $ \begin{pmatrix} 1.0 & 0.2 & 0.2 \\ 0.2 & 1.0 & 0.2 \\ 0.2 & 0.2 & ...
0
votes
0answers
21 views

Suppose $f_{X,Y}(x,y) = \frac{3(4-2x-y)}{16}$ for $x>0$, $y>0$ and $2x+y<4$. Find $P(Y>2\mid X=1/2)$

Suppose $f_{X,Y}(x,y) = \frac{3(4-2x-y)}{16}$ for $x>0$, $y>0$ and $2x+y<4$. Find $P(Y>2\mid X=1/2).$ Not sure if I have the correct solution. Would someone let me know if this makes ...
1
vote
1answer
19 views

simple poisson application

I am trying to solve a previous exam question for one of my courses, but there are no solutions.I've tried to somehow link this to the uniform distribution since we know exactly how many tourists ...
2
votes
0answers
12 views

Additivity of cumulants of dependent random variables?

What sequences of real-valued random variables $X_1,X_2,X_3,\ldots$ exist for which for all $n$ and all $k$ $$ \operatorname{cum}_k (X_1+\cdots+X_n) = \operatorname{cum}_k(X_1)+\cdots + ...
3
votes
1answer
57 views

What does it mean that log-normal distribution is positively skewed?

I'm writing an economic overview and I need to get an explanation in the context of log-normal distribution being derived from the idea of multiplicative influence of factors and in order to explain ...
1
vote
0answers
16 views

Confirming the triangular inequality for Lévy-metric $d_L(F,G)$ and $d_L(F,G)<\infty$

Let $F,G$ be cumulative distribution functions. The Lévy-metric is defined to be $$ d_L(F,G)=\inf\left\{h\geq 0: F(x-h)-h)\leq G(x)\leq F(x+h)+h~\forall x\in\mathbb{R}\right\} $$ I would like to ...
0
votes
0answers
16 views

Continuity points of cumulative distribution function

Let $F_n,n\geq 1$ and $F$ cumulative distribution functions and assume that for all $x$ and for arbitrary $\varepsilon >0$, if $n\geq N$, we have $$ F_n(x-2\varepsilon)-2\varepsilon\leq F(x)\leq ...
-1
votes
1answer
36 views

Probability Mass Function of having both loaded & fair coins [closed]

Suppose a box contains many coins that are either biased (loaded) or balanced. A loaded coin has probability of landing on its head as p ∈ (0.5, 1.0), and a balanced coin, of course has probability ...
0
votes
1answer
36 views

Arg min E(X-b)^2

Question is that Find argmin E(X-b)^2 Where X is a continuous random variable. I think the minimum of E(X-b)^2 is 0. Because (X-b)^2 is nonnegative. But how can i find argmin E(X-b)^2 ? And can i ...
1
vote
1answer
25 views

Maximum of standard brownian motion on an interval

I'm trying to find the probability that the maximum of standard Brownian motion on the interval $(t_1, t_2)$ exceeds a value $x$, i.e., $$P(max_{t_1 \le s \le t_2}B(s) \gt x)$$ I initially ...
0
votes
1answer
36 views

Conditioning Poisson on Poisson [closed]

There is a bus, whose departure at a stop is distributed as poisson(mu). People arriving to the same stop is distributed as poisson(lambda). Find the PMF of the number, N, of people on any given bus. ...
1
vote
1answer
34 views

When is $\nabla^2 f (x, y, z)= $ probability measure?

When is $\nabla^2 f(x, y, z) $= probability density function ? That is $\nabla^2 f(x, y, z)= \mu (x, y, z)$ $\int \mu (x, y, z) dxdydy = 1 $ What conditions must $f(x,y,z) $ satisfy? It is known to ...
0
votes
1answer
28 views

derive the pdf for “difference of log-normal distributions”

Can someone please help me to derive pdf for $X$, $$ X = \frac{\ln(f_1) - \ln(f_2)}{b_2-b_1} $$ here $f_1$ and $f_2$ are normal distributions with different means and standard deviations, and $b_1$ ...
3
votes
2answers
40 views

Why can we 'choose' continuity points?

Let $F$ and $F_n$ be distribution functions with $\lim_n F_n(x)=F(x)$ for all continuity points $x$ of $F$. In a proof there is the following part: Block quote [...] choose the finite points ...
0
votes
0answers
19 views

Negative Utility Function

I'm using a negative utility function to compute portfolio allocation, $u(x) = -p_1e^{-X/T} + -p_2e^{-Y/T} + -p_3e^{-Z/T}$ where $p_1 + p_2 + p_3 = 1$ Certainty equivalence of this I get through, ...
0
votes
1answer
75 views

Distribution of a process dependent on a Markov chain's states

Consider a Markov chain $X_t$ with state space $\{0,1\}$, initial distribution $$ \begin{array}{l} \mathbf{P}(X_0=1)=\lambda \\ \mathbf{P}(X_0=0)=1-\lambda \end{array} $$ and transition ...
2
votes
0answers
21 views

Deriving $(n-1)\dfrac{S^2}{\sigma^2} \sim \chi^2(n-1)$ [duplicate]

I can accept the fact that $Z^2 = \dfrac{\left(X-\mu\right)^2}{\sigma^2} \sim \chi^2(1)$ without knowing too much about this mysterious $\chi$-function, but I'm wondering how I can show that ...
2
votes
1answer
30 views

Distribution Function is absolutely continuous or singular?

$$F(x) = \begin{cases} 0,& x < -1\\ \frac{x}{3} + \frac13,& -1 \leq x \leq 0\\ \frac{x}{3} + \frac23,& 0 < x \leq 1\\ 1,& 1 \leq x \end{cases}$$ This $F(x)$ is a distribution ...
0
votes
0answers
14 views

Lagrangian method with objective function and constraints in expected value form.

Im reading a paper and over last two weeks I have been involved with a mathematical calculation. It is about maximizing the principal utility under uncertainty; max $\int G(x-s(x))f(x,a)dx $ , where ...
0
votes
1answer
24 views

Meaning of symbol “$y\nearrow x$” in CDF Limit

Could somebody explain the meaning of "$y\nearrow x$"? $F_X$ is right continuous, that is, for any $x$, $\lim_{y \nearrow x} F_X (y) = F_X(x)$.
0
votes
1answer
15 views

Question about the support of a joint distribution

Let X and Y be continuous random variables having the joint pdf $$f(x,y) = 8xy , 0\leq{y}\leq{x}\leq{1}$$ Find $g(x|y=\frac{1}{2})$ the conditional pdf of $X$ given $Y = \frac{1}{2}$ I found that ...
0
votes
0answers
28 views

Probability with ordered statistics and exponential distribution involved

Assume that $X_1,X_2$ are independent random variables with exponential distribution with the same mean 100. Let $X_{(1)}=\min\{X_1,X_2\}$ and $X_{(2)}=\max\{X_1,X_2\}$. Calculate ...
0
votes
0answers
19 views

Distribution function of Sum of IID Exponentiation Variables of Variable amount

So I'm trying to determine the distribution function of a random variable, S, give: $N \sim Geo(\frac{1}{1+\lambda}) $ $S_i \sim Exp(\mu), \forall i\in [0,N]$ $S = \Sigma^{N}_{i=0}S_i$ $S = ...
1
vote
1answer
28 views

Find the conditional pmf of $Y$ given $X = 0$

Let $X$ and $Y$ have the joint pmf defined by $f(0, 0) = f(1, 2) = 0.3$, $f(0, 1) = f(1, 1) =0.2$ $(a)$ Tabulate the conditional pmf of $Y$ given $X=0$ $(b)$ Tabulate the conditional pmf of $X$ ...
0
votes
0answers
28 views

joint pdf for two independent uniform distribution

Suppose that $𝑋_1$ and $𝑋_2$ are independent and follow a uniform distribution over $[0, 1]$. Let $𝑌_1 = 𝑋_1 + 𝑋_2$, and $𝑌_2 = 𝑋_2 − 𝑋_1$. a) Find the joint pdf $𝑓_{𝑌_1,𝑌_2} (𝑦_1, 𝑦_2)$ ...
1
vote
1answer
27 views

Finding the Probability of a Normal Distribution

The mean IQ scores of 30 primary school students is 108.56 and the Standard deviation is 12.33. Assume that IQ scores for primary school students that have been kept for 50 years illustrate a normal ...
3
votes
1answer
41 views

Finding a joint probability mass function

I have to find the joint probability mass function (pmf) of (X,Y) for the following problem: Roll a die repeatedly until a five or six appears, and let X be the number of rolls before a five or six ...
1
vote
2answers
31 views

Method for separating 'randomness' and 'non-randomness'

Let's assume I have a random two signals: Sin(t) R(t) Sin(t) is of course the trignometric function, but R(t) is a random process. So let's now assume I ...
0
votes
1answer
87 views

$N = Poisson(\lambda)$, $\{U_i\}$ iid $\implies (N_1, N_2) = Po(\lambda p_1)$x $Po(\lambda p_2)$

Let $\{N\}\cup\{U_i\}$ be independent random variables. $N = $ Poisson$(\lambda)$ $\{U_i\}$ iid, taking values in $\{1,2\}$, $\mathbb{P}[U_i = 1] = p_1$ and $\mathbb{P}[U_i = 2] = p_2$, $p_1 + p_2 ...
0
votes
1answer
25 views

Cumulative distribution function (CDF) strictly less than

Suppose a distribution function for the random variable $X$ is given by $$F(x)=\left\{ \begin{array}{11} \hfill 0 \hfill & x \lt 0\\ \hfill \dfrac{x}{2} \hfill & 0 \leq x \lt 1\\ \hfill ...
1
vote
0answers
19 views

Generate Correlated Normals

I want to generate normals $X,Y,Z$ with the correlation matrix $R$ but with means $0, 1, 2$ and variances $4, 16, 25$ respectively. How can I do this? Is it possible to apply Cholesky?
-2
votes
1answer
34 views

How do can i solve the integral, finding cdf [closed]

Let $X$ be an exponential random variable with mean 1 and Y a uniform random variable between $0$ and $1$. Assume X and Y are independent and let $Z =e^{X/2}$ Find the joint cumulative ...
1
vote
0answers
14 views

Generating 'bursty' traffic using probability distributions beyond Poisson

I'm trying to develop a more realistic vehicle generation model for populating a traffic microsimulator. I'm trying to model a real-world intersection from which I have historical flow data. Currently ...
1
vote
1answer
30 views

Find $g(x|y=\frac{1}{2})$, the conditional pdf of $X$ given $Y = \frac{1}{2}$ (Need confirmation)

Let X and Y be continuous random variables having the joint pdf $$f(x,y) = 8xy , 0\leq{y}\leq{x}\leq{1}$$ I found that the marginal pdf of Y is $f_2(y) = 4y - 4y^3$. Does $g(x|y=\frac{1}{2}) = ...