1
vote
1answer
19 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
0answers
29 views

Simplification of Double Integral with Independent Parameters

I am trying to find a posterior distribution and the hint is that the double integral in the denominator should simplify because $p1$ and $p2$ are independent. $\displaystyle \int$$\displaystyle ...
1
vote
1answer
51 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\;\;\;\; x<2\\ \dfrac{(x-2)^2}{3}+0.3\;\;\;\;\;\;\;2\leq x < ...
1
vote
1answer
36 views

Finding the mean with absolute value

This question is out of my field and topic that I am teaching myself now, but I was wondering how would you solve this problem if it had the absolute value of it. My Question: $$f(x) = ...
3
votes
1answer
25 views

Maximum likelihood to throw exactly two 6s

One throws a dice $n$ times. For which value of $n$ is maximum the probability to obtain exactly two 6s? I get $$n=11 \text{ or } n=12.$$ My solution: the probability to obtain exactly two 6s in ...
0
votes
1answer
19 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 ...
1
vote
1answer
30 views

Weird question about probability density function

I'm assuming "actual" means the total probability of the PDF (the integral from $-\infty to \infty$) must be 1, so $$\int\limits_{-\infty}^{\infty} ke^{-0.1t}dt = 1$$ Wolfram Alpha seems to be ...
0
votes
0answers
34 views

Integrals as continuous sums

I came a cross this paragraph while reading a book on probability: "The length of a subset $S$ of $[0,1]$ is the integral $\int_S dt$". Unfortunately I am unable to understand this part. Can anyone ...
0
votes
1answer
35 views

Double Integration with interesting variable limits, and difficult function

I am trying to reconstruct the following probabilistic model, \begin{equation} \begin{split} \frac{1}{\mu}\int^{\infty}_{0}P(N \geq n\, |\, L=l, T=t)\,e^{-\frac{l}{\mu}} dl &= ...
1
vote
2answers
74 views

Using L'hopital's rule to solve problem.

Show that $$\lim_{x \to 0} \frac{-3x }{e^{x/3}}=0 $$ by L'hopital's rule. I know how to solve this without using L'hopital's rule. I was just reading about this and was wondering can we solve it ...
0
votes
0answers
54 views

MMSE estimate for scalar gaussian & uniform prior

I am trying to analyze the behavior of an MMSE estimator given Guassian measurement with scalar variability on an underlying uniform prior distribution. The measurement is generated according to the ...
0
votes
1answer
39 views

Specify a function that majorizes $\frac{2}{\pi}\sqrt{1-x^2}, -1\leq x \leq 1$ [closed]

Specify a function that majorizes $\frac{2}{\pi}\sqrt{1-x^2}, -1\leq x \leq 1$ Could anyone please help me? I dont have a clue how to start.
0
votes
2answers
34 views

Memorylessness and its square

If we have that $T$ is a memoryless random variable, how do we know if $T^2$ is one too? I am supposed to investigate the cases $T: \Omega \rightarrow \mathbb{R_{\ge 0}}$ and $T: \Omega \rightarrow ...
0
votes
1answer
17 views

Different ways to give cards

The question is: You have 48 cards and want to distribute them to 4 players(so everyone gets 12). How many ways are there to do so? My idea was to take 4 cards in each step and distribute them to the ...
1
vote
0answers
29 views

expected value with integration

For the exponential distribution, $f(x)=(1/\theta) e^{-x/\theta}$ for $x>0,$ and $f(x)=0$ for $x \leq0$ $(i)$ Determine the exact value for the probability $P(0<X<3\theta).$ I need help ...
1
vote
1answer
29 views

Integration related to exponential random variable

Given $\lambda,T>0$, consider the following integral. \begin{align} X&=\int_0^T t\lambda e^{-\lambda t}\mathrm d t= \left[ -t e^{-\lambda t} \right]\Big|_0^T +\int_0^T e^{-\lambda t} \mathrm ...
1
vote
1answer
20 views

Animal jumping equally likely to the left and right.

We have an animal that starts at the point zero and jumps equally likely to the left(-1) and right(+1). After 2k jumps, where $k \in \mathbb{N}$ ,it arrives back again at the point $0$. The question ...
2
votes
1answer
75 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 = ...
0
votes
0answers
25 views

Is there any version of Jensen's inequality for quasiconvex function

I am looking for some generalization of Jensen's inequality for functions $g:\mathbb{R}^n \rightarrow \mathbb{R}$ where $g(x)$ is quasiconvex (or not convex). We known that for convex functions, ...
2
votes
2answers
55 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
58 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
0answers
28 views

Proof convolution formula two stochastic variables

Let's say I have two continuous independent stochastic variables, defined on $(0, \infty)$. With densities: $X_1$ ~ $f_1(t_1), t_1 \in (0, \infty)$ $X_2$ ~ $f_2(t_2), t_2 \in (0, \infty)$ The ...
1
vote
0answers
26 views

Battery between liftimes

Suppose that the operating lifetime of a certain type of battery is an exponential random variable with $\theta$ $= 2$ (measured in years). Find the probability that a battery of this type will have ...
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 ...
0
votes
0answers
20 views

How to understand this equation for brownian motion

I am reading this article from the notes 'an intro to SDE'. Here I dont know why in (1) he take that integral from - infinity to infinity. I mean why we do that? I just dont know what the physics or ...
1
vote
1answer
23 views

choosing the function of a random variable with the lowest variance w.r.t.o the mean of that random variable?

Consider a real gaussian random variable with mean $\theta$ and unit variance. Let $y$ be an observation of the random variable. The objective is to estimate $\theta$ over all possible $y$. Let ...
0
votes
2answers
67 views

Intuitive idea behind the probability density function

as an application of Calculus, I am currently teaching some material about continuous random variables. My main example is the height $X$ of a French male chosen randomly in the French population. ...
1
vote
1answer
31 views

Random variable of a store

The weekly profit in thousands of dollars of Miller's Office Supply Store is random variable X whose cdf is given as follows: $F(x)=0$ for $x<0$; $F(x)=(3/32)(2x^2-x^3/3)$ for $0 \leq x \leq 4$; ...
0
votes
1answer
46 views

Independence random variables

I found two theorems in my notes and they seem to be somewhat complementary which made me doubt that both of them are true: a) Let $X,Y: \Omega \rightarrow \mathbb{R}$ be a measurable function and ...
1
vote
1answer
28 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
55 views

Derivative of Expected value

Say we have a random variable $X$, with density function $f(x)$, and moment generating function $M(t) = E[e^{tX}]$, and we take the derivative - $M(t) = \frac{d}{dt}E[e^{tX}] = E[\frac{d}{dt}e^{tX}]$ ...
0
votes
2answers
221 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
31 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 ...
0
votes
1answer
23 views

Which probability in this hypothesis test?

We have a hypothesis A (null hypothesis) such that $p\le 0.6$ and B such that $p>0.6$. Now we want to develop a deterministic test $\phi$ for 20 people that has a safety of 95%. Hence we would be ...
0
votes
0answers
75 views

If $X_1, X_2$ have exponential distributions what distribution does $Y = \sqrt{X_1^2 + X_2^2}$ have?

If $P_{X_1}(x) = P_{X_2}(x) = k \exp(-k x)$ how will $Y = \sqrt{X_1^2 + X_2^2}$ be distributed? $X_1$ and $X_2$ are independent. What I have Done: The distribution for $Y=X_i^2$ must be ...
1
vote
1answer
43 views

Compute expectation (Ito integral/calculus)

I am having trouble computing this expectation. Does anyone know how to proceed? $$E\left[e^{2B(t)} \int_0^t s dB(s) \right].$$ Is it 0? I tried expressing $e^{2B(t)}=1+ 2\int_0^t ...
3
votes
3answers
66 views

Upper Bound for $p(1-p) $ where $0 \le p \le 1$

In a book on statistics, I've seen the upper bound of $p(1-p) $ to be $$ p(1-p) \le \frac{1}{4} $$ for $0 \le p \le 1$ which seemed correct. I tried to duplicate this with a simple derivation, ...
3
votes
1answer
82 views

Integration and theorems on continuous functions

$f(x)$ is positive and continuous function on $\mathbb{R}$ and, moreover, $\int_{-\infty}^{+\infty}f(x)dx=1$. $\alpha\in(0;1)$ and $[a;b]$ is the interval having a minimum length such that the ...
0
votes
2answers
50 views

how to compute this expectation value

A random variable $X \sim N(0,1)$, compute $\Bbb E(X^n)$ . I manage to do this by characteristic function. Now I try to compute this by moment generating function or do it directly. So I have 2 ...
1
vote
2answers
44 views

References for probability using Calculus

I have to teach a Calculus class (details on the syllabus below) and I want to add some applications to other sciences. But I would like to avoid the classical physics examples, because the physics ...
0
votes
1answer
21 views

Prob $n$-th arrival of Poisson Process ($\lambda_{1}$) occurs before $m$-th arrival of P.P. ($\lambda_{2}$)?

Define $ T_{1} $: time until $n$-th arrival of $ \{N_{1}(t)\} $; $T_{2} $: time until $m$-th arrival of $ \{N_{2}(t)\} $. $ \{N_{1}(t)\}, \{N_{2}(t)\} $ independent. Then \begin{align} ...
0
votes
3answers
41 views

How to calculate the summation of this sequence

$$\sum_{j=0}^n {n \choose j}e^{iuj}p^j(1-p)^{n-j}$$ Here, $i$ stand for complex number $i$, $j \in N$, and $0<p<1$. Since it is a sequence, I coundn't find any formula for this. The answer is ...
0
votes
2answers
45 views

How to calculate that series

I was looking at the solution of a problem, then this: I don't know how to caluculate that series in the denominator, and here I assume the result is done by write out that series. Here X is a ...
0
votes
0answers
101 views

Suppose that Z has a standard normal distribution and that Y is an independent $\chi^2$-distributed…

Suppose that Z has a standard normal distribution and that Y is an independent $\chi^2$-distributed random variable with v df. Then $T=\frac{Z}{\sqrt{\frac{Y}{v}}}$. If Z has a standard normal ...
1
vote
1answer
73 views

Help with an integral for: If U has a $\chi^2$ distribution with v df, find E(U) and V(U)

If U has a $\chi^2$ distribution with v df, find E(U) and V(U). By definition, $E(U) =\int^{\infty}_{0} u\frac{1}{\gamma(\frac{v}{2})2^\frac{v}{2}}u^{\frac{v}{2}-1} e^\frac{-u}{2}\,du ...
5
votes
3answers
72 views

Probability with Calculus

A point is chosen randomly in the region bounded by the curve $y = x^2$ and the line $y = 4$. Find the probability that the $y$-coordinate is less than $a$ for any $a$ in[0,4]. I think that I need to ...
1
vote
1answer
34 views

Convergence condition of infinite cosine product

Please show that, given that $\sum_{k\ge1}c_k^2=\infty$ and $c_k\rightarrow 0$, $$\lim_{n\rightarrow\infty}\prod_{k=1}^n\cos{tc_k}=0$$ for every $t\neq0$. (All variables here are real numbers.) The ...
1
vote
2answers
38 views

Calculate the value of c for which f is a probability density.

Let f the function defined by: Where c is positive none zero and constant . How can i calculate the value of c for which f is a probability density.Thnxs for the help.
0
votes
0answers
34 views

System of integral equations for a unimodal symmetric probability distribution

Let $f(x)$ be a symmetric unimodal probability distribution on $\mathbb R$, with mean $\mu=0$. By unimodal, I mean that $f(x)$ is strictly increasing for $x<\mu$ and strictly decreasing for ...
3
votes
1answer
478 views

Is this infinite sum always less than zero?(+500pts bounty for the correct answer)

I wonder if the following infinite sum is always negative for all (finite) $A,d>0$ and $B<0$. Any counterexample also suffice. Here is the sum: $$\frac{\partial}{\partial d}\sum_{n=1}^\infty n ...