0
votes
1answer
29 views

Finding an expression for a joint probability if two random variables have the same distribution function.

If $X$ and $Y$ are independent random variables with the same distribution function, say $F$, find an expression for $P(X<t, Y<t)$. My attempt: $P(X<t, Y<t) = P(X<t)P(Y<t) = ...
2
votes
0answers
42 views

Algebra problem involving Q functions

I have the following algebra problem, which is actually the end-part of my bigger research problem. Let $a$, $b$ and $m$ be reals with $a<b$. Also, let $Q(\cdot)$ denote the Gaussian ...
1
vote
1answer
29 views

Find the cumulative distribution function.

$f(x)=20 x (1-x)^3$ over $0 \le x \le 1$ and $0$ elsewhere. I know that by definition the cumulative distribution function if $F(x) = P(X \le x) = \int_{-\infty}^x f(t)\,dt$. In this case, I must ...
0
votes
0answers
27 views

Two quick questions regarding joint densities and joint mass functions.

Given: $f(x,y)=c(x+y)$ at $(1,2), (2,1), (1,3), (3,3)$ and zero elsewhere. My work: I found $c=\dfrac{1}{16}$ by solving $1 = \int_{-\infty}^\infty \int_{-\infty}^\infty f(x,y)\,dy\,dx$. Question 1: ...
0
votes
1answer
15 views

Finding the Baye's Estimator

The likelihood for a single observation is $f(x|\theta)=\frac{1}{\theta}$ and the prior distribution is a $gamma(4, \lambda)$ so the distribution is $\frac{\lambda^{4}}{6}\theta^{3}e^{\lambda\theta}$ ...
1
vote
1answer
57 views

Summation of: $\sum_{1}^{\infty}\left(\frac{2}{3}\right)^x$ [duplicate]

This is a subsection in my statistics homework. It goes back to calculus II and summations, and it's been a long time since I've studied it so I'm rusty. I'm looking to solve the summation of ...
4
votes
3answers
65 views

Deriving Mean and Variance of Laplace Distribution

It has been a long time since I have used calculus, and I am trying to understand how the mean and variance of the Laplace distribution with pdf $$f(x|\mu,\sigma) = \dfrac{1}{2 ...
2
votes
2answers
77 views

Log normal distribution - Where am I wrong?

Let $X$ be a R.V whose pdf is given by $$f(x)=p\frac{1}{\sqrt{2\pi\sigma_1^2}}\exp\left(-\frac{(x-\mu_1)^2}{2\sigma_1^2}\right)+ ...
1
vote
2answers
47 views

Integration by parts

Integrate using integration by parts: $F(y) = (y+1)e^{-y}$ Find: Evaluate the $\int_{a=0}^{b=\infty}F(y)\;dy$ using integration by parts. Thus far, I've distributed the $e^y$ term and split ...
1
vote
1answer
19 views

Simplify $\dfrac {\partial }{\partial b}\left( \sigma ^{2}\right) =0$ and $\dfrac {\partial }{\partial m}\left( \sigma ^{2}\right) =0$

Given $\sigma ^{2}=\sum ^{N}_{i=1}\left[ y_{i}-\left( mx_{i}+b\right) \right] ^{2}$ How to simplify $\dfrac {\partial }{\partial b}\left( \sigma ^{2}\right) =0$ and $\dfrac {\partial }{\partial ...
2
votes
1answer
31 views

Rearranging equation with algebra

I'm having a difficult time showing that the two are equivalent: $2(x_1-\theta)(1+(x_2-\theta)^2)+2(x_2-\theta)(1+(x_1-\theta)^2) = 2(\bar{x}-\theta)(1+(x_1-\theta)(x_2-\theta))$ I have multiplied ...
0
votes
2answers
32 views

Probability that the call will be answered at time $t$ is given by $f(t)$. Find the median waiting time for the call.

$$f(t) = \begin{cases} 0 & \text{if $t < 0$ } \\ 0.2e^{-t/5} & \text{if $t\geq 0$} \end{cases}$$. $ $ Find the median waiting time for the call. $ $ I cannot understand ...
0
votes
4answers
63 views

Min and Max of $ f(x,y)=\frac{x-y}{a-x-y}$

I'd like to find max and min of $$ f(x,y)=\frac{x-y}{a-x-y}$$ where $0\le x<y\le a/2$. Any one can suggest? Thank you
0
votes
1answer
24 views

Calculating optimum values of $u$ and $m$ from $\mathbb V(\bar {y_2}\prime)=\frac{S_2^2(n-u\rho^2)}{n^2-u^2\rho^2}$

I have to find optimum sample size in sampling on two occasions. Suppose that the samples are of the same size n on both occasions. In selecting the second sample, $m$ of the units in the first ...
0
votes
1answer
39 views

Derivative of an exponentially weighted moving average

It has been a while since my university math courses, so let me apologize right off the bat... I'm using GSL to perform non-linear regression analysis and am mostly happy with the outcome, however, ...
0
votes
1answer
20 views

Infimum of Gamma distribution

Let $X$ be a Gamma random variable with the CDF $F_X(x)=\frac{1}{\Gamma(\alpha)}\gamma(\alpha,\beta x)$ where $\Gamma(x)$ represent the gamma function and $\gamma(a,b)$ denotes the lower-incomplete ...
1
vote
1answer
62 views

Just learned about the bell curve in statistics. How is calculus related to this curve?

I'm learning about the bell curve in statistics and I'm trying to understand the calculus behind the concept. I've taken calc 1 already. How is the integral related to this ...
0
votes
1answer
21 views

What is the joint cumulative distribution function of two separate uniformly distributions ~[0,1]?

I was asked to find a joint distribution for two suppliers where each alone has a uniformly distributed demand~U[0,1], how do I do that? In general is it possible to join in general any two non ...
1
vote
3answers
59 views

Simplify function with polynomial via least-squares

I want to "adjust" (simplify) $f(x)$, a function, by $g(x)$, a polynomial, via least-squares. I want to write code for that. Apperently my code is issuing wrong results, so I was wondering if my ...
1
vote
1answer
66 views

Evenly spreading study over 20 days?

Want to study $8$ hours a day, and $4$ hours on exam days. Want to study all exams exactly the same amount of time total. Exams in $10$,$13$,$19$,$20$ days from start. What is the daily ...
1
vote
1answer
27 views

Finding the marginal density of $ce^{-x^2-y^2-xy-x}$

In my probability class, we've started studying joint distrubutions and I've been tasked with the following problem: Let $(X,Y)$ have joint density $ce^{-x^2-y^2-xy-x}$, where $c>0$ is some ...
2
votes
1answer
82 views

Is Calculus a requirement for Better Statistics?

Is Calculus really required to be better at Statistics and Probability and to be a good Data Science person? Arthur Benjamin Says "Very few people actually use calculus in a conscious, meaningful ...
0
votes
1answer
74 views

Partition a set into n subsets with elements with sum equal to m

For example, given the set $a_n = \frac{n}{20}$ for $\,n \in \{1,\,2,\,\dots,\,19\}$, I would like to get all possible partitions of this set in 4 subsets such that the sum of their elements is always ...
4
votes
2answers
67 views

Approximating the erf function

I was trying to find an approximate solution to the following: $\DeclareMathOperator\erf{erf}$ $$\frac12 \sqrt{\pi} \erf\left (\frac{x-2}{\sqrt{10}}\right) + \frac12 \sqrt{\pi} \erf ...
0
votes
1answer
17 views

Finding the Marginal of $f_y(x,y)$ from the density function

So I have the following function $f(x,y)=10xy^2$ and I am asked to find the marginal of $y$ with the region being $0<x<y<1$. This is my setup: $f_y=\int_y^1 10xy^2 dx$ This is correct right ...
1
vote
1answer
54 views

$Pr(X+Y \geq \frac{\pi}{2})$

I want to find $Pr(X+Y \geq \frac{\pi}{2})$ for joint pdf $f_{X,Y}(x,y) = x \cos y, 0 \lt x \lt \frac{\pi}{2}, 0 \lt y \lt x, 0$ otherwise. I believe I have found conditional pdf of $Y$ given $X=x$ ...
2
votes
1answer
55 views

Finding the conditional pdf of $Y$ given $X=x$ from a joint pdf. Answer confirmation!

I have a continuous joint pdf, and I am working out the conditional pdf of Y given X=x. Is my method correct? I am given: $f_{X,Y}(x,y) = x\cos y , 0 \lt x \lt \frac{\pi}{2}, 0 \lt y \lt x$ ...
0
votes
0answers
56 views

Deriving the halflife of a Ornstein Uhlenbeck mean reverting process

Given the O-U process $$ dy_t = (hy_{t-1}+m)dt + dW_t $$ where $m$ is the mean, $h$ is the regression coefficient and $W_t$ is the Wiener process. how would does one derive that the half life is ...
1
vote
1answer
107 views

How to evaluate $\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n-1}{2}\right)}$

How to evaluate $$\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n-1}{2}\right)}$$, where n is integer > 0? I know the gamma function formula will give $$ ...
0
votes
2answers
63 views

Painful? Moment Generating Function

Part 1 Let $X$ be a random variable with the p.d.f. $f(x)=\frac{1}{4\pi}e^{\frac{-x^2}{4}}$, compute the MGF of $X$. So I know I want ...
1
vote
1answer
20 views

Calculate the Marginal Probability

f($X_1$, $X_2$| $p_1$, $p_2$) = $p_1^{x_1}$(1-$p_1)^{({n_1}-{x_1})}$$p_2^{x_2}$(1-$p_2)^{({n_2}-{x_2})}$ $p_1$~Unif(0,1) independently $p_2$~Unif(0,1) $n_1$=34 $n_2$=56 Calculate the marginal ...
0
votes
1answer
13 views

Integrate a PDF over a set

This is part of a proof of Chebyshev's inequality, but there's one line I'm just trying to clarify my intuition of: $f(x)$ is the pdf and $g(x)$ is a non-negative function. $$ \int\limits_{x: g(x) ...
1
vote
1answer
60 views

Two Methods of computing E[X] but I get 2 different answers instead of the same

The 1st method is $\int_{A}^{B}xf(x) dx$ and the 2nd method is $A+\int_{A}^{B} 1-F(X)$ I have the following CDF $$F(X)=\begin{cases} 0\qquad x<2\\ \dfrac{(x-2)^2}{3}+0.3\qquad 2\leq x < 3\\1 ...
0
votes
1answer
27 views

Limit containing inverse normal

I am having trouble taking the following limit that contains an inverse normal distribution as alpha approaches 1: $\lim_{\alpha \to 1} \frac{\mu + \sigma \frac{\phi^{-1}(\alpha)}{1-\alpha}}{\mu + ...
15
votes
2answers
2k views

Average IQ of Mensa

I was wondering, what the average IQ at Mensa is. Mensa is a group of people with an IQ of at least 130. And the IQ is normally distribed with $\mu = 100$ and $\sigma = 15$. My idea was this: To ...
0
votes
1answer
25 views

sum of two normal distributed random variables

Consider $Z = \frac{1}{2}(X+Y)$ where X and Y are normal distributed variables. It is easy to show that the variance of $Z$ is $1/2$ and the mean is zero. I'm stuck because I want to show this using ...
0
votes
1answer
24 views

Joint continuous random variable conditional probablity

Okay so here is the problem: The PDF is $$f(x,y)=6(1-x)\;\; 0\leq y \leq x \leq 1$$ The question asks find $P(0<Y<0.25\;|\;X=0.5)$. I approached the problem like this ...
0
votes
0answers
7 views

Calculating population for significance

I am trying to rewrite the standardnormal function so I can calculate for which population n, the test is significant. z= (c1-c2)/(f1-f2)^0.5 f1+f2= (c1(1-c1))/n + (c2(1-c2))/n This is in need for ...
0
votes
1answer
47 views

Double Integration with interesting variable limits, and difficult function

I am trying to reconstruct a probabilistic model, I have tried different methods of approach, by parts, substitution, but to no avail. Any help with this would be greatly appreciated!
2
votes
1answer
84 views

Differentiate $P_{x_n}(z) = \prod_{i=1}^n\frac{1+z+z^2+…+z^{i-1}}{i}$ twice to calculate the variance of involutions.

Use the Probability Generating Function for Involutions: $P_{x_n}(z) = \prod_{i=1}^n\frac{1+z+z^2+...+z^{i-1}}{i}$ To Calculate the Variance of Involutions where: $Variance \space X_n = ...
1
vote
0answers
61 views

Proof about lognormal distribution

I'm trying to prove a result about the lognormal distribution that seems to me to be fairly intuitive, but I can't get the proof to work. Basically, I'd like to prove that as the mean increases, the ...
2
votes
2answers
67 views

Poisson distribution proof question

I'm reading over the Poisson distribution proof and trying to understand how $$\frac{n(n-1)\cdots(n-k+1)}{(n-\lambda)(n-\lambda)\cdots(n-\lambda)}$$ tends to 1 as $$n\rightarrow\infty\text{ ?}$$ ...
3
votes
2answers
60 views

Poisson distrubution proof question.

I was reading over the proof for the Poisson distribution and came across this sentence: "But since $$\left[1-\frac{\lambda}{n}\right]^n\rightarrow e^{-\lambda}$$ as $$n\rightarrow\infty$$, ..." Can ...
0
votes
1answer
47 views

Calculating mean vector of a multivariate distribution

I have a question concerning calculating the mean vector (vector of expected values) of a general multivariate distribution. I try to obtain the mean vector by doing a vector integration and I ...
1
vote
2answers
115 views

maximum likelihood - estimate $\sigma^2$ in $N(0,\sigma^2)$

Prove, using maximum likelihood, the estimation of $\sigma^2$ where $X$ is $N(0,\sigma^2)$ There is no real statistics involved, just algebra and finding partial derivatives, so I tagged it algebra ...
1
vote
1answer
34 views

Expectation of a Uniform PDF

How do I find the expectation of the following pdf? $f(x,y) = 1/\pi r^2$ , where $x^2+y^2 \leq r^2$ I've tried to integrate it on the bounds $-\sqrt{1-x^2}$ and $\sqrt{1-x^2}$ for $\int ...
1
vote
1answer
35 views

Unbiased estimate $\lambda^2$

Given a Poisson distribution I want to figure out whether $d:(x_1,...,x_n) \mapsto x_1^2$ and $d':(x_1,...,x_n) \mapsto x_1x_2$ are unbiased estimations for $\lambda^2$ ? I mean it would sound ...
0
votes
1answer
90 views

Proof that a median minimizes 1-norm. [duplicate]

I was wondering whether there is an easy way to show the following: We have a data set $x_1,...,x_n$ and $m$ is a median if for at least half of the n data points we have that $x_i \le m$ and for ...
0
votes
2answers
324 views

Version 2:Help finding the probability that $Ax^2 + Bx + C$ has real roots?

Suppose that $A, B,$ and $C$ are independent random variables, each being uniformly distributed over $(0,1)$. What is the probability that $AX^2 + BX + C$ has real roots? I am given a hint that if ...
0
votes
1answer
33 views

Check for Independence

Given $$f_{(U_1,U_2)}(u_1,u_2)=\begin{cases} 1/2& -u_1<u_2<u_1 \text{ and } u_1 - 2 < u_2 < 2 - u_1 \text{ and } 0 < u_1 <2\\ 0& \text{otherwise}\end{cases}$$ I found that ...