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

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1answer
22 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
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0answers
7 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
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1answer
24 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
11 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
37 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$(\lambda)$. We let U = X + Y and V = Y, and the ...
0
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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
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1answer
29 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
16 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 ...
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2answers
37 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
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1answer
24 views

Product of two distribution functions.

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

Normal distrubition [on hold]

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 ...
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0answers
17 views

Deriving the Pareto Distribution from an Exponential Distribution [on hold]

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 ...
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votes
1answer
21 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
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0answers
45 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 ...
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votes
0answers
9 views

What is the normal distribution probability? [on hold]

1) A company is contemplating surveying the passengers on a particular ferry service. Over the years, the average number of passengers per trip on the ferry service has been 60 and the standard ...
0
votes
2answers
18 views

Conditional probability in multinomial distribution

Consider a multinomial distribution with $r$ different outcomes, where the $i$th outcome having the probability $p_i$, $i$=1,...,$r$, $\sum_{i=1}^r p_i = 1$. Denote $X_i$ be the number of times the ...
2
votes
1answer
31 views

Using the inverse Gaussian integral to find percentiles

I need some help with the following: Let $$R=\mu+\sigma*\epsilon \hspace{1cm} \epsilon \sim N(0,1)$$ I want to argue that $$ \mu + \sigma*\Phi^{-1}(u)$$ are the percentiles of the model when ...
0
votes
3answers
55 views

Is it even possible to find the variance of this moment generating function?

This is my moment generating function: $M_x(t) = \frac{6e^t}{t^2} + \frac{6}{t^2} + \frac{12e^t}{t} - \frac{12e^t}{t^3} + \frac{12}{t^3}$. I have to find the mean the variance of it. After taking ...
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1answer
54 views

What is the probability the best case occurs? (Comp Sci Type Question)

I'm having trouble figuring out what's the probability the best case occurs? It's my first time bringing together probabilistic knowledge into computer science. The question goes as such. Consider ...
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1answer
33 views

An inverse problem on tail probability

This is a question out of curiosity. Assume that $f(x)$ is a density function for which there is a constant $C>0$ so that $$ \int_t^\infty f(x) dx \le C f(t) $$ holds for large enough $t>0$. My ...
3
votes
1answer
20 views

Generating random variables with complicated probability distribution functions

I have an interesting question I need to solve, and as much as I try, I cannot wrap my head around it. Given a postive random variable X with p.d.f. ...
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2answers
34 views

Let $X_{1},X_{2}, \dots, \sim Exp(1)$ i.i.d. - Calculate the probability of $P[\max{(X_{1},\dots,X_{n},)} < \log(n) - 5] $ for $ n > e^{5}$

Let $X_{1},X_{2}, \dots, \sim Exp(1)$ i.i.d. - Calculate the probability of $P[\max{(X_{1},\dots,X_{n},)} < \log(n)-5] $ for $ n > e^{5}$ as well as $n \rightarrow \infty $ The correct ...
0
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1answer
18 views

For exponential random variables $X_i$, how to find $P(t-X_1<X_2\mid t-X_1<X_3)$?

Assume $X_1, X_2, X_3$ are three independent exponential random variables with means $1/A$, $1/B$ and $1/C$ resp. How do we calculate $P(t-X_1<X_2\mid t-X_1<X_3)$? My try: \begin{align} ...
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0answers
24 views

Stat problem! Why is this? [duplicate]

This is a statistics problem. although this is not a problem which needs an answer, I want to know the reason Why this is right. Can you guys help me ? Thanks in advance!
1
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1answer
30 views

Marginal PMF values of a function

I've been doing this question and I was wondering if my workings are correct, if they are not correct, can you please correct them? The question is as follows: My workings are: $\binom{y}{x} ...
0
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0answers
17 views

Generalised chi square PDF: 2 normal DOF with different widths

I'm trying to derive the PDF $f_Z(x)$ for the random variable $Z = X^2 + Y^2$ where $X \sim N(0, \sigma_1)$ and $Y \sim N(0, \sigma_2)$. Searching here I've found this solution, which is only valid if ...
2
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0answers
35 views

How can this be derived??

This is a statistics problem. although this is not a problem which needs an answer, I want to know the reason Why this is right. Can you guys help me? A deck of $n=10$ cards is numbered from 1 to ...
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0answers
18 views

What type of distribution can be used to describe a game with positive expected winnings?

I've come across something I'm not too sure about. Let's say we flip a coin, heads mean we lose 1 unit, tails means we win a 1 unit. This distribution of outcomes in this would be considered normal, ...
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0answers
11 views

Exercise on Variance Reduction Techniques

I'm struggling with a variance reduction techniques exercise of a book I'm reading. Anyone has an idea of how to approach this problem? Can anyone help me out?
1
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1answer
25 views

Is there a shorter way for me to answer this joint probability question?

Suppose a box contains 10 green, 10 red, and 10 black balls. We draw 10 balls from the box by sampling with replacement. Let X be the number of green balls, and Y be the number of black balls in the ...
3
votes
2answers
46 views

Random variable $X^2$ determined by moments

Let $X$ be a real random variable, with standard normal distribution. Is the distribution of $X^2$ determined by its moments? In general, if $n \in \mathbb N$, is the distribution of $X^n$ ...
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0answers
21 views

Exchangeability and pairwise exchangeability

Suppose we have $\{Y_{i}\}_{i=1}^{n}$ that is exchangeable so the joint distribution of this sequence is the same as $\{Y_{\sigma(i)}\}_{i=1}^{n}$, where $\sigma:\{1,...,n\}\rightarrow\{1,...,n\}.$ ...
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votes
1answer
87 views

How do you answer this Bayes theorem question? [on hold]

Your computer is acting strangely and you suspect it has a virus. Unfortunately all 5 of your virus detection programs are outdated. If your computer has a virus then each program, independently of ...
0
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0answers
17 views

Simulating beta random variables

Let's say I estimated the parameters of Beta distribution with a single-period dataset, and want to generate a multi-period sample. How can I do that? To be more specific, I estimate my parameters ...
1
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1answer
21 views

Finding the CDF of $g(X)$ where $X$ is a continuous random variable

I imagine this is a rather simple problem, but I'm having a bit of a hard time actually finding the answer. $X \sim \mathrm{Exp}(0.2)$ and $W=g(X)$ given by $g(X) = \begin{cases} X^{\frac{1}{3}} ...
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0answers
33 views

Joint density of $X_1^2+X_2^2\ \text{and} X_2,\ X_i\sim N(0,1)$

Let $X_1 $ and $X_2$ be iid with a common standard normal distribution. I am looking to find the joint pdf of $Y_1 =X_1^2 +X_2^2$ and $Y_2=X_2$. I know i can use a straight transformation argument but ...
0
votes
1answer
22 views

Derivative of integral over part of Gaussian distribution

I am currently trying to compute the following derivative and integral: $$ P\psi_\theta = \frac{d}{d\theta}\int_{-k}^k tf_T(t)dt, $$ where $t=x-\theta$ and $X\sim N(\theta_0,\sigma^2)$. $f_T$ above ...
0
votes
1answer
27 views

Bivariate distribution question

How would I do this question by hand? I know I integrate from $-\infty$ to $\infty$ for $f_{x,y}$, but I have no idea how to do it by hand! My algebra soup is bad, can someone please help me? P.S I ...
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0answers
43 views

Given that X has known PMF , find the PMF of Y [duplicate]

The whole question is: Let $X$ be a discrete random variable and let $Y = 0.5 X + 3$. (i) Assume that the PMF of $X$ is given by where k is some suitable constant. Determine the value of k. (ii) ...
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1answer
12 views

Two random variables X,Y are, X,Y independant b. are X+Y X-Y independant

if X and Y are independent, check whether the (0, 0) value is the same as P(X=0) P(Y= 0), and the same with the other 4 entries. Make a table with the distributions of X + Y and X - Y. For any ...
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2answers
49 views

Defining a probability density function in R (software), and sampling from it

I'm asked to generate a random sample from a logistic distribution with the PDF, $$f(x)=\frac{e^{-x}}{(1+e^{-x})^2}$$ without using the function rlogis(...) So I ...
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0answers
15 views

Mean of Binomial Distribution

I'm having a little trouble with this, and I'll admit my knowledge in this topic is weak. I working through some practice problems in a physics books, but this is a math problem so I thought I'd post ...
0
votes
1answer
22 views

Variance from the pdf

FOr the interpretation of the variance, it is the fluctuation of the data around the mean. So if I know that mean (say mean=0), and then there are lots of data (70%) points are greater than +/-10 away ...
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0answers
32 views

How to prove that the distribution function Fx is left continuous if and only if the distribution law µ is non atomic [on hold]

How to prove that the distribution function $F_x$ is left continuous if and only if the distribution law $\mu$ is non atomic. Can the law $\mu$ and lebesgue measure be singular if the distribution ...
1
vote
1answer
65 views

Explanation of how probability density functions transform under the change of variable

I've just read about probability density function from this article. In that article, there is some wired concept that I can't understand, please see the section named "Dependent variables and change ...
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votes
1answer
30 views

Question I couldn't identify to solve this distribution [on hold]

A couple decides to have 3 children.If none of them is a girl,they will try again,and if they still don't get a girl,they will try again and continues so on.If X is the number of children,the couple ...
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votes
4answers
30 views

Proving consequence of $\operatorname{var}(X)+\operatorname{var}(Y)=\operatorname{var}(X+Y)$

How to prove that if $\operatorname{var}(X)+\operatorname{var}(Y)=\operatorname{var}(X+Y)$, then $X$ and $Y$ are independent?