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

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0
votes
2answers
557 views

Looking for the logic of a sequence from convolution of probability distributions

I am trying to detect a pattern in the followin sequence from convolution of a probability distribution (removing the scaling constant $\frac{6 \sqrt{3}}{\pi }$: ...
0
votes
1answer
28 views

Probability of sample mean [on hold]

A town has $500$ real estate agents. The mean value of the properties sold in a year by these agents is $\$800,000$ and the standard deviation is $\$300,000$. A random sample of $100$ agents is ...
1
vote
0answers
27 views

Apparently same probability questions with different answers.

I was reading A first course in probability by Sheldon Ross when and then I came up with this question. This is how he introduces the famous problem of points Independent trails, resulting in a ...
0
votes
0answers
28 views

Probability and Expected profit

I really need help for this qn! You are asked to determine the profitability of a new line of sunglasses, which will retail for \$10. The fixed cost of setting up the line is \$2000. The total number ...
0
votes
2answers
24 views

Flipping several biased coins

Assuming I'm flipping $M$ biased coins with different probability for heads $p_i, i=\{1,...,M\}$. What is the probability of having $k$ times head? Is there a distribution function known for this?
2
votes
2answers
34 views

Finding $f_Y$ such that $Z=Y\cos(X)\sim\mathcal{N}_{0,\sigma}$ for $X\sim\mathcal{U}[0,2\pi]$

I need to choose the probability distribution $f_Y(y)$ of a random variable $Y$ such that the variable $Z=Y\cos(X)$ is normally distributed with zero mean, i.e. ...
-1
votes
0answers
23 views

Distribution of the supremum of a transformed Brownian Motion?

I have a stochastic process given by $z_{t}=w_{t}/\alpha\left(t\right)$ , where $w_{t}$ follows a Wiener process (a standard $\left(0,1\right)$ Brownian Motion) starting from $w_{0}=0$ , ...
0
votes
0answers
9 views

Try to prove that the discrete distribution function is a singular distribution function

Actually it is the 6th question in the Section 3 of Chapter 1 in the book named The Course in Probability Theory by Chun Kai Lai. Someone asserts that the derivate of the discrete function on the ...
0
votes
2answers
32 views

How can I do a constructive proof of this:

Say Z is a non-negative R.V, and P(Z>0)>0. Then exists a a>0 and an b>0 such P(Z>a)>b. I am not sure how to start with the proof, I've been assigning numbers than can qualify for some CDFs but I don´t ...
-1
votes
0answers
8 views

Probability Scheme of Variable X [on hold]

from digits 1,2,.....9 we are making 3-digit number. Variable X is measuring number of 1-digits in this number. Write probability scheme for varibable X?
0
votes
0answers
39 views

Make the sum of random variables converge, while the sum of the variances diverges

Suppose $X_n$, $n=1,2,3,...$, are independent and $Var(X_n)$ is uniformly bounded by finite constant $C>0$. Construct $X_n$ such that $\sum_nX_n$ converges a.s., but $\sum_nVar(X_n)=\infty$.
-1
votes
0answers
16 views

Stat problem help me! [on hold]

Hello! I got a problem when I was solving stat problem. I solved by(c), but after that, I found it hard to solve. Can you guys help me or give me a hint? Thank you anyway!
1
vote
1answer
28 views

The distribution of sample proportion for given population proportion and sample size

If the population proportion is 0.90 and a sample of size 64 is taken, what is the probability that the sample proportion is more than 0.89? (4dp) work: $n=64$, $\hat p=0.89$, so $X=n \hat p ...
0
votes
1answer
32 views

Pdf of the product of an exponential r.v. and a beta r.v.

Let $X$ and $Y$ are 2 independent random variables, where $X$ has an exponential distribution with parameter $1$ and $Y$ is $\beta(a,b)$ distributed. What is the Pdf of $W=XY$ ? Thanks !
1
vote
1answer
27 views

Ratio between normal distributed and gamma distributed variables

Let $X \sim N(0,1)$ and $G \sim Gamma(a)$. Why is $\frac{X}{G}$ t-distributed?
1
vote
1answer
22 views

A urine test, the VMA test

Neuroblastoma is a rare, serious, but treatable disease. A urine test, the VMA test, has been developed that gives a positive diagnosis in about 70 % of cases of neuroblastoma. It has been proposed ...
0
votes
0answers
26 views

approximation by binomial [on hold]

Consider a finite region $\Lambda \subset \mathbb{R^2}$ (e.g. a square), and a model in which $N$ indistinguishable particles are placed randomly and uniformly in $\Lambda$. Let $\Delta \subset ...
0
votes
0answers
9 views

average dirichlet distribution [on hold]

Is it possible to combining 2 Dirichlet distribution averaging their values? The resulting probability distribution is still a Dirichlet distribution? If not how can I merge 2 similar Dirichlet ...
0
votes
1answer
20 views

How to compute $P(X\leq Y)$ and $E(X^2 Y)$ with given probabilities [on hold]

Given $P(X=1, Y=0) = 0.1; P(X=1, Y=1) = 0.1; P(X=1, Y=2) = 0$ and $P(X=2, Y=0) = 0.2; P(X=2, Y=1) = 0.4; P(X=2, Y=2) = 0.2$ How do I compute $P(X \leq Y)$ and $E(X^2 \cdot Y)$?
1
vote
0answers
30 views

Expectation of absolute value of Brownian motion

I'm working on this problem that I can't seem to figure out. The problem involves a 1-dimensional Brownian motion, $B_t$, where the subscript denotes the time, and it asks me to show that the ...
0
votes
1answer
30 views

Relationship between a distribution function and the truncated distribution function

Let $F(x)$ be a distribution function and $G(x)$ be $F(x)$ truncated on some interval $(a,b)$. I want to show that: $$G(x)=\frac{F(x)-F(a)}{F(b)-F(a)}, a<x \leq b$$ I want to do this by using ...
-3
votes
0answers
52 views

CDF of two random variables [on hold]

Suppose Y and Z are independent N(0, 1) random variables and suppose X = |Z|. Consider the random point (X, Y). (a) Derive the CDF FD(d) = P(D ≤ d) of the distance from the origin D =√X2 + Y2. Sketch ...
-1
votes
1answer
36 views

some question about expected value

Let $X$ be a non negative random variable. We Know that $E(X)=0$. Is that correct that $X=0$ for some $X$. And more general: Is there a point in the probability space for which E[X]≤X and a ...
0
votes
2answers
107 views

How to find $P(X>x)$ when the density is known but the integral does not seem to converge

I am trying to evaluate $$P(X>x) = \int_x^{\infty } t^{\kappa } \exp{\left(-\rho t^{\alpha\kappa + 1}\right)} \, dt$$ where $\kappa$, $\rho$ and $\alpha$ are all constants. I have tried some ...
-2
votes
0answers
25 views

function of a random variable problem 2 [on hold]

Let $U$ be a continuous random variable with uniform distribution over $[0,1]$. Define $X$ by $$X=\operatorname{Int}\left(\frac{\ln(1-U)}{\ln(1-p)}\right)+1$$ where $\operatorname{Int}(x)$ is the ...
0
votes
2answers
28 views

Ratio of Gamma random variables

If $X_i$, $i=1,2$ are independent gamma$(\alpha_i,1)$ random variables, find the distribution of $\frac{X_1}{X_1+X_2}$ and $\frac{X_2}{X_1+X_2}$. Attempt: Let $Y_1 = \frac{X_1}{X_1+X_2}$ and ...
1
vote
1answer
21 views

Finding distribution of random variable if X is exponential $(1)$

Let X be an exponential (1) random variable, and define Y to be the integer part of X+1, that is $\hspace{15mm}Y=i+1$ if and only if $\hspace{5mm}i \leq X \leq i+1, i = 0,1,2,...$. Find the ...
3
votes
1answer
44 views

$P(X^2+Y^2<1)$ of two independent n(0,1) random variables

Suppose that X and Y are independent n(0,1) random variables. a) Find $P(X^2+Y^2<1)$ Attempt: a) Let $U = X^2 + Y^2$, $V = Y$. Then $X = \sqrt{V^2 -U}$, $Y = V$. $J = \left| ...
0
votes
1answer
25 views

Uniformity of the difference between two random variables

What can I say about the distribution of two random variables $A$ and $B$ such that $A-B$ is uniformly distributed?
1
vote
1answer
42 views

Pdf of the product of an exponential rv and a $f_Y=Ka^{-K}y^{K-1}$ distributed rv …

Let $X$ and $Y$ are 2 independent random variables, where $X$ has an exponential distribution with parameter $1$ and $Y$ has the following Pdf: $f_Y=Ka^{-K}y^{K-1}, 0 \le y \le a $. Someone claims ...
0
votes
0answers
10 views

Exponential Generalised Beta distribution type 2

I am doing some toy examples with EGB2. I would like to simulate variates and then estimate back parameters I used for the simulation. I can achieve the former by using gamlss.dist package in R. ...
1
vote
0answers
23 views

Correlation and First Order Stochastic Dominance

Suppose we have a random variable $X \sim [0,1]$ with a continuous distribution $F_X(x)$. Suppose $I \in \left\{0,1\right\}$ is a discrete random variable with $\text{Prob}(I=1 \ | \ X=x)$ strictly ...
0
votes
1answer
11 views

Percentages in Normal Distribution

A statistics problem involves: Lengths of a certain type of carrot have a normal distribution with mean 14.2 cm and standard deviation 3.6 cm. (i) 8% of carrots are shorter than c cm. Find the value ...
-2
votes
1answer
25 views

A drug treatment [on hold]

A certain drug treatment cures 90 % of cases of hookworm in children. Suppose that 20 children suffering from hookworm are to be treated, and that the children can be regarded as a random sample from ...
1
vote
0answers
8 views

Creating random integers with distribution schema

I need to create an array that includes 0..5 integers. I'm able to create them randomly. But I need to create them according to below distributions. How can I get below distributions? Ps: I'm using ...
0
votes
1answer
29 views

Explicit CDF associated to Gamma PDF [on hold]

Thanks in advance for the help with this! I'm struggling to follow the solution in the book for this problem. Any help is greatly appreciated. Let the distribution function of X for x>0 be: $$F(x) = ...
0
votes
1answer
12 views

Binomial probabilities

Okay, so here is probably the easiest question ever on this website. A question on binomial distribution. In a city, the percentage of left-handed women is 16% and the percentage of left-handed men ...
0
votes
2answers
51 views

Poisson random variables and Binomial Theorem

I'm working on a problem from Casella and Berger's Statistical Inference. X is distributed as Poisson$(\theta)$ and Y is distributed as Poisson$(\lambda)$, with X and Y being independent. We let U = X ...
0
votes
0answers
7 views

Deriving the multivariate t-distribution from the normal mixture representation

I'm trying to derive multivariate t-distribution from its representation as a normal variance mixture distribution by following the calculations in Appendix 4 of ...
0
votes
1answer
34 views

Estimating how much two probability distributions differ

I have two probability distributions A and B. First I would like to estimate how much they differ. In this regard I use as metric the Jensen–Shannon distance (i.e. the square root of Jensen–Shannon ...
1
vote
1answer
18 views

How to visualize probability distributions in terms of sets - joint and marginal?

Let there be two sets, $\mathcal{X},\mathcal{Y}$, both finite, and they represent the set of values that the discrete random variables, $X,Y$ can take. $\mathcal{P}_{Y|X}$ be all possible ...
1
vote
2answers
41 views

Find $E[N]$, where $N = \min\{n>0: X_n = X_0\}$

Let $X_i$, $i\geq 0$ be independent and identically distributed random variables with probability mass function $$ p(j) = P\{X_i=j\},\; j=1,...,m,\;\sum^{m}_{j=1}P(j)=1 $$ Find $E[N]$, where ...
1
vote
1answer
27 views

Product of two distribution functions.

Let F and G be two distribution functions, does the product FG still a distribution function?
0
votes
0answers
7 views

Uniform conditional on maximum distribution

If $U_1,U_2,\dots,U_n$ are i.i.d. $U(0,1)$ and $U_{(n)}=max(U_1,U_2,\dots,U_n)$, I want to show that $U_n|U_{(n)}$~$U(0,U_{(n)})$. I know that the pdf of $U_{(n)}$ at $t$ is $nt^{n-1}$. I did the ...
-1
votes
0answers
35 views

Normal distrubition [closed]

Let Xi denote the weight of a randomly selected prepackaged one-kilogram bag of potatoes. Of course, one-kilogram bags of potatoes won’t weigh exactly one kilogram. Actually, history suggests that Xi ...
-2
votes
0answers
21 views

Deriving the Pareto Distribution from an Exponential Distribution [closed]

Let $T$ be an exponential random variable with hazard rate $a>1$. Consider a random variable defined by the condition $X = b (e^t - 1)$." I need to find the density of $x$.The answer is ...
-2
votes
1answer
22 views

Get unknown value in discrete random variable

Let $X$ be a discrete random variable (i) Assume that the PMF of $X$ is given by $$\operatorname{Pr}(X=x)=\begin{cases}kx^{2} & x \in \{-4,-2,0,2,4\} \\ 0 & x\not\in \{-4, -2, 0, 2, ...
2
votes
1answer
33 views

Expected Value on code

I'm trying to figure out the expected number of times this algorithm will print. I'm stuck on how to go about doing so. I used an indicator variable to keep track of the number of print statements ...
2
votes
0answers
53 views

Autocorrelation function of random process

Let $X_t$ be a wide sense stationary random process indexed by $t\in\mathbb{R}$ with finite mean and variance. (http://en.wikipedia.org/wiki/Stationary_process) Q1) Is the autocorrelation function ...
0
votes
1answer
11 views

maximum-likelihood: a sequence of events described by a Bernoulli distribution

I am having quite some troubles with the following homework: In a city it's measured for the whole year whether it rained or not. A distribution $\textrm{Bernoulli}(r_t|\rho)$ characterizes the ...