Questions tagged [probability-distributions]

Questions on using, finding, or otherwise relating to probability distributions, probability density functions (pdfs), cumulative distribution functions (cdfs), or other related functions.

Filter by
Sorted by
Tagged with
-1
votes
0answers
14 views

we use the poisson random distribution

Let ܺ denote the number of typographical errors on a single page of the lecture notes. Let’s assume that ܺ has Poisson distribution with parameter ߣ = 1. Calculate the probability that there is at ...
0
votes
1answer
18 views

how can we derive both the CDF of $M$ and the PDF of $M$

Given $X_i \sim Uniform[0, 1]$ for $i = 1, \dots, n$. What is the distribution of $M := \min(X_1, \dots, X_n)$? I feel like I'm missing something and I've been stuck on this for two days
0
votes
0answers
21 views

Probability of difference between any two random variables

Given random variables $x_1,x_2,\ldots,x_n$ which are i.i.d. with $x_i \sim \mathcal N(\eta, \sigma^2), \forall 1 \leq i \leq n$. I tried to find the probability that none of these random variables ...
0
votes
1answer
15 views

A general expression for the sum of multiple independent Normal Mixture Distributions?

Suppose random variable $X_1$ is a mixture of two Normal distributions with means of $\mu_A$ and $\mu_B$ respectively, standard deviations of $\sigma_A$ and $\sigma_B$ respectively, and weights given ...
0
votes
0answers
10 views

Relating Total Variation Distance to Number of Samples

I have a dataset whose underlying joint distribution is $P_{XY}$. $X$ is the input and $Y$ is the output. I restrict someone to take $n$ samples from this dataset. Based on the samples he got, he ...
0
votes
1answer
14 views

Determining the distribution and showing it's a pivot

so we let $X_1....X_n$ be iid random variables from the exp($\theta$) distribution. Find the distribution of S=$2\theta\sum_{i=1}^n(X_i)$ and hence show that S is a pivot. Now i understand that the ...
1
vote
1answer
16 views

Do these conditions imply weak convergence of the random variable?

Here is a question from a past exam from probability theory that I try to tackle: Let $X,X_1,X_2,\ldots$ be real random variables. We know that: (a) $X_n^2$ converges in distribution to $X^2$ (b) $...
0
votes
1answer
22 views

Expected value of the longest run of heads or tails in N flips of a coin

What is the expected value of the longest run of heads or tails observed in $N$ flips of a fair coin? I'd like to consider this for any $N$, small or large.
0
votes
2answers
13 views

What is the most probable number of acceptable screens in the next batch of 10 screens and what is the probability?

A novel process of manufacturing laptop screens is under test. In recent tests, it is found that 75% of the screens are acceptable. What is the most probable number of acceptable screens in the next ...
0
votes
2answers
25 views

Normally distributed rain drops problem

About 50% of raindrops land downtown, downtown is a perfectly circular space around the city centre. Assuming the coordinates of the raindrops are independent and distributed according to the standard ...
0
votes
1answer
10 views

Calculating Variance of sample variance

How do you find the variance of sample variance, without using moment generating functions but using the fact that variance of a chi squared distribution with n degrees of freedom, is equal to 2n? ...
0
votes
2answers
25 views

Likelihood function for MLE

Refer to the Example 7 in this lecture: How did the author obtain the likelihood function ? Is it from binomial? Can someone show the steps to the likelihood function? Thank you!
7
votes
0answers
68 views

Is this distribution already known and has a name?

I am asking whether the distribution on $\Bbb R$ with probability density $$ f(x) := \frac 2 {\sqrt{2\pi}} e^{-\frac{x^2}{2}} - 2 \vert x\vert \int_{\vert x \vert}^\infty \frac 1{\sqrt{2\pi}} e^{-\...
2
votes
1answer
24 views

Roll $m$ dice and re-roll up to $n$ of them where the result is less than $t$. What is the probability that a single die will have the value $v$?

I'm trying to combine some dice probabilities into one. If I roll $m$ dice, and after check each die individually against some threshold $t$. I can then re-roll up to $n$ of them if the result of ...
0
votes
1answer
42 views

Convolution of two uniform

Let $X$ be a uniform random variable on $[0,1]$, let $Y$ be uniform in $[3,5]$ independent of $X$. Find the probability density function of $X + Y$. My solution is: $$ (f_X * f_Y)(x) = \begin{cases} ...
0
votes
1answer
12 views

Poisson process show $E(T_1|N_1=0)=1+\frac{1}{\lambda}$

For a Poisson process with rate $\lambda$, show $E(T_1|N_1=0)=1+\frac{1}{\lambda}$. attempt We know $T_1 \sim Exp(\lambda)$. The solution I'm looking at says to use the memoryless property. I don'...
0
votes
0answers
23 views

Calculate number of defects(PMF of sum of multinomial distributions)

Suppose you have two factories, A and B which produce two items at the same time with a probability of defect from A, $p_A$ and probability of defect from B, $p_B$ Suppose a total of $N_T$ items are ...
2
votes
2answers
42 views

General solutions of $x+y+z=x^2+y^2+z^2=x^3+y^3+z^3$

Note: this actually has nothing to do with probability theory. Fell free to skip ahead and solve the equations below. Since I am unable to solve this continuous version of a problem about moments, I ...
0
votes
0answers
15 views

Does there exist a probability distribution such that all central moments are equal?

I find here that if the moment generating function of a random variable has positive radius of convergence, then that random variable is determined by its moments. So does there exist a continuous ...
1
vote
0answers
24 views

Binomial Distribution Trends [closed]

I have to find the probability of exactly 0, 1, 2 and 3 blue bricks being in the different rows. I've already done this and posted the image of the completed table of probabilities. My question is ...
0
votes
0answers
65 views

Fair coin flip problem

Given a fair coin which is flipped $n$ times in succession, and let $X_t$ be the number of heads received among the first $t$ flips. Let $m < n$. I need to calculate the probability that there ...
-1
votes
0answers
6 views

computing the expectation of a softmax function

Would anyone suggest what is the expectation of the following softmax function \begin{equation} P(\mathbf{s}_{t}=i|\mathbf{s}_{t-1}=j,\mathbf{u}_{t-1}=k,\mathbf{x}_t, \mathbf{R}) = \frac{\exp\...
1
vote
0answers
43 views

Convolution exponential uniform. Can anyone help please?

Let $X$ be a uniform random variable on $[0,1]$, let Y be an exponential random variable with parameter $\lambda$ independent of $X$. Find the probability density function of $X + Y$ I know that ...
-1
votes
0answers
10 views

Random variable X on $\Omega$ =[0,1] [closed]

Let $\Omega$ =[0,1] $\\$ $ X: \Omega \rightarrow \mathbb{R} $ s.t $ X(\omega)=min (\omega ,1-\omega) \forall \omega \in \Omega $. Can someone calculate : $ X^{-1} (]-\infty,x ])$
0
votes
0answers
19 views

X random variable on [0,1] [closed]

Let $\Omega$ =[0,1] $\\$ $ X: \Omega \rightarrow \mathbb{R} $ s.t $ X(\omega)=min (\omega ,1-\omega) \forall \omega \in \Omega $. Can someone calculate : $ X^{-1} (]-\infty,x ])$
0
votes
0answers
19 views

Deconstructing a pdf into two independent pdfs

I would appreciate any help or suggestions with the decomposition. In words, I'm asking when a random variable, that contains information about two other random variables, can be deconstructed as a ...
1
vote
0answers
42 views

Limiting distribution about Poisson

Let $X_1,X_2...X_n \sim U(0,1)$ be $i.i.d $, $S_n=\sum_{m=1}^{n}X_m$, please find the limiting distribution of $\sum_{m=1}^{n} \mathbb I_{X_m S_n\leq1}$. I guess that might be Poisson distribution. ...
1
vote
1answer
15 views

The Fast Fourier Transform of this Kernel Blows Up

Warning: This type of math is new to me, and I am self learning for my thesis, so I am coming at this question with a shakey understanding to begin with. I am trying to take the Fast Fourier ...
0
votes
1answer
12 views

Truncated distribution formula proof

This definition is taken from the Wikipedia page for truncated distribution: Let $X$ be a random variable with a continuous distribution, $f(x)$ be its probability density function and $F(x)$ be its ...
0
votes
1answer
21 views

Probability of being in a random interval

Consider an Erlang$(n,\lambda)$-distributed random variable $A$ and another Erlang$(1,1)$-distributed random variable $B$ (hence exponential(1)) such that $A$ and $B$ are independent. How to compute $...
0
votes
0answers
24 views

$X\sim\text{Unif}[-1,2]$ find PDF of $Y=X^2$

$X\sim\text{Unif}[-1,2]$ find PDF of $Y=X^2$ Would someone mind explaining to me the parts of the solution I'm confused about? "Since $-1\le X\le2$ and we have $0\le X^2\le4$" I understand the 4 ...
0
votes
1answer
13 views

Expected value of truncated distribution

This definition is taken from the Wikipedia page for truncated distribution: Let $X$ be a random variable with a continuous distribution, $f(x)$ be its probability density function and $F(x)$ be its ...
0
votes
0answers
18 views

Does exist other techniques for finding PDF/CDF from relation between random variables?

I know two techniques of finding CDF/PDF from relation between random variables. By relation I mean that one rv is represented by other, examples $ Y = X^2 $ or $ Z = \frac {U} {1-U} $. Summarizing, ...
0
votes
1answer
30 views

How to see a function is Gaussian

I have this function $$f(x, y) = \frac {1}{2\pi}\exp(−0.5(x^2-2xy+9y^2))$$ I proceed like this: First I compute $\Sigma^{-1}$ which is \begin{bmatrix} 1 & -1 \\ -1& 9\end{bmatrix} Then I ...
0
votes
1answer
18 views

Covariance Formula

Hy everybody, I have $f_X(x)$=$(3/14)(x^2+2)$ and $f_Y(y)$=$3/28((1/3)+y)$ and the joint $f(xy)$=$(3/56)(x^2+y)$ I have to compute the $Cov(XY)$, I proceed like this: I know they are dependent, ...
0
votes
0answers
25 views

Derive the t-distribution using transformation of random variables.

I'm asked to derive the pdf of the t-distribution following way. $$V = \frac{Z}{\sqrt{U/n}}$$ where $Z$ is the standard normal and $U$ is the chi-squared distribution with degree of freedom n. ...
1
vote
0answers
13 views

Topic-prevalence in Latent Dirichlet Allocation

In Focused Topic Models, the main motivation is to decouple the global topic prevalence and in-document prevalence. However, I couldn't see how the original Latent Dirichlet Allocation (LDA) couples ...
0
votes
2answers
40 views

Coin flipping game - expected value

Given a weighted coin that lands on heads 55% of the time, you flip the coin until you get you get your first tails. For each heads, you make \$1. When you flip tails you lose \$0.90. What is the ...
0
votes
0answers
17 views

$A, B, C$ $IID$ $geometric(p);$ $E[\frac{A - 2B + C}{A}]$ [duplicate]

I know how to do all the steps but I am stuck when calculating $E[\frac{1}{A}]$. Any properties of the geometric distribution (i.e apart from solving using the pmf) to solve this? And for people who ...
0
votes
2answers
27 views

conditional expectation of uniform distribution given the realization of product of uniform distributions

Assume that $A \sim U(0,1)$, and $B \sim U(0,b)$ with $b<1$, and A and B are independent. Can we calculate (closed form for) the expected value of $A$ given that we observe the realization of AB, ...
0
votes
1answer
16 views

Sequences of probability distributions which do not converege uniformly and satisfy integral properites.

I'm looking for two (convergent) sequences of real-valued functions, $\{f_{k}\}$ and $\{g_{k}\}$, such that, for each $k$, $$\int_{-\infty}^{x}f_{k}(t)dt\leq \int_{-\infty}^{x}g_{k}(t)dt$$ at every $x$...
1
vote
1answer
31 views

In a MLE what are the rules for getting things out of the product symbol $\prod$ in our $f(x_1,…,x_n | \theta )$?

In a MLE what are the rules for getting things out of the product symbol in our $f(x_1,...,x_n | \theta )$? What I mean is once we have the density function of our distribution we write $f(x_1,...,...
1
vote
1answer
29 views

Forgetting about the Underlying Probability Space

I have read that when dealing with random variables, one often forgets about the underlying probability space (Wikipeida). What is a good example (or a couple) of when one does this? I figure that ...
0
votes
0answers
23 views

Pretesting-squeezing for a distribution

Suppose we have: $$ f_X(x) \propto \left\{ \begin{array}{ll} \frac{1}{(x-a)(b-x)}\mbox{exp} \left\{ -\frac{1}{c}(d+\mbox{log}(\frac{x-a}{b-x}))^2 \right\}\quad & a<x<b \\ 0 \quad & \mbox{...
0
votes
0answers
8 views

Any probability distribution for simple $F(x|\theta)/f(x|\theta)$?

I would like to find one probability distribution satisfying following properties. (1) The conditional pdf is $f(x|\theta)$, $x\in [-\overline{x},\overline{x}]$, increases for $x\in [-\overline{x},\...
0
votes
0answers
20 views

Compute probability of event involving two independently distributed random variables, one normal, one uniform

I am looking to compute the probability of an event involving two independent random variables, X and Y, where $X∼N(2700,300)$ and $Y∼U(0,400)$ I am looking for $Pr(3000+Y<X)$. I have searched ...
0
votes
1answer
25 views

Convergence in distribution of this random variable [closed]

Let $(X_n)_{n\in \mathbb N}$ indipendent random variables such that: P($X_n$ $\in \mathbb {1,-n^2}) = 1$ and $E[X_n]=-1$ for each $n$. It follows that: $P(X_n=1) = (n^2-1)/(n^2+1)$, $P(X_n=-n^2) = ...
1
vote
1answer
21 views

Prove independence of limit point of a sequence of random variables

Let $X_1, X_2, \ldots$ be a sequence of random variables on a probability space $(\Omega, \mathcal{F}, P)$, and let $\mathcal{G}$ be a $\sigma$-subalgebra of $\mathcal{F}$. Assume $X_n$ is ...
0
votes
0answers
19 views

Weak convergence of product of random variables [duplicate]

Suppose $X_n$ and $Y_n$ are random variables defined on the same probability space. Prove that if $X_n$ converges weakly to a random variable $X$ and $Y_n$ converges weakly to a constant $c$, then ...
2
votes
2answers
43 views

Let $X\sim\mathrm{Normal}(1,4)$. If $Y=0.5^X$, find $\Bbb E[Y^2]$.

Let $X\sim\mathrm{Normal}(1,4)$. If $Y=0.5^X$, find $\Bbb E[Y^2]$. So I have $\Bbb E[Y^2]=M^{\prime\prime}_Y(0)$ So I need to compute $M_Y(t)=M_{0.5^X}(t)=\Bbb E\left[e^{0.5^X}\right]$ $$\Bbb E\...