This tag refers to questions about functions and measures of continuous variables, such as real and complex variables. This tag is in contrast with discrete variables, such as integers and enumerable sets.
Questions tagged [continuous-variables]
35 questions
5
votes
2 answers
Is the Lambert W function the Newton flow of the exponential function?
Is this right?
The Lambert W function, denoted by $W(z)$, is defined as the inverse function of $f(z) = ze^z$. In other words, if $w = W(z)$, then we have $z = w e^w$.
The continuous Newton's method is a technique for finding the roots of a function…
user1057203
3
votes
1 answer
Simple algebra of integral
I'm studying economics and I'm having trouble with math calculations.
In my economics textbook, the following equation comes out, and for me it's hard to understand how this relationship is established.
I know a very basic concept of calculus, so…
guest
- 143
- 5
3
votes
2 answers
Statistics, PDF and CDF pre-university (A level)
This is a question from an A level textbook on continuous random variables. It states that the CRV $T$ has pdf $f(t)=0.5 ~for~ 1
2
votes
1 answer
Expectation when a partial range is known
Let $X$ be a continuous random variable with a density $f_X(x)$ and let $A\subseteq \mathbb{R}$. Would it be true to claim that:
$E[X | X\in A] = \int_{x\in A} x \cdot f_X(x) \cdot dx$?
My understanding of dependent expectation is that:
$E[X|X\in A]…
Eric_
- 995
2
votes
1 answer
Why does the Wiener process use $\sqrt{dt}$ instead of $dt$? Why does simulation of random walk in continuous-time not occur as expected?
I'm trying to make a point about Markov processes and ran into difficulty understanding how to simulate continuous-time random walks. A discrete-state discrete-time random walk has the equation:
$x_{n+1}=x_n+X_1,\qquad X_1 \in \{-1,1\}$,
while the…
Coletti
- 33
2
votes
0 answers
Equvivalent definitions of mutual information of continuous random variables
I am reading Elements of Information Theory by Cover and Thomas (2006) and struggle with the definition of mutual information for continuous random variables (Chapter 9: Differential Entropy). For two random varibles with a joint pfd $f(x, y)$, they…
Avec
- 21
2
votes
1 answer
How to squeeze the logistic function obliquely?
The original function: $f(x) = \frac{1}{1+e^{-10(x-0.5)}} $. Its graph (blue line) is shown here:
How can I squeeze this function obliquely along the $y=x$ line? The squeezed function $g$ needs to satisfy that $g(0.5) = 0.5$, $g(0)$ is close to…
Dong Cao-Huu
- 31
2
votes
0 answers
Can someone help me to prove the orthogonality in continuous case for Bessel Functions?
I already prove the orthogonality condition of Bessel functions for discrete case ($0,b$).
$$\int_0^{b}\rho J_{\nu}(\chi_{\nu l}\rho/b)J_{\nu}(\chi_{\nu l'}\rho/b)d\rho = \frac{b^2}{2}[J_{\nu+1}(\chi_{\nu l})]^2\delta_{ll'}$$
Now, I need to prove…
User13122015
- 29
2
votes
0 answers
A question on i.i.d CRVs ~ Uniform$([0, 1])$
If $X_1$, $X_2$, and $X_3$ are Continuous Random Variables that are i.i.d Uniform$([0, 1])$, what can deduce about $P(X_1 \le X_2)$ and $P(X_1 \le X_2 \le X_3)$?
My original thought is that $P(X_1 \le X_2) = 1/2$ because they are identically and…
Anony mouse
- 21
- 3
1
vote
1 answer
Intersecting Intervals in a Uniform (0,1) distribution
Five intervals are each selected according to the following procedure: two points are sampled from
U[0,1], the larger becoming the right endpoint and the smaller becoming the left endpoint. What is the probability that there exists a point of…
user1657506
- 63
- 4
1
vote
1 answer
Should discrete and continuous time models give the same results?
Today I thought a lot about very simple population models, and there are still a few things that bug me.
Consider a simple discrete exponential growth function:
$$ n(t+1) = n(t) + n(t) b - n(t) d = n (1+r) $$
And recursively expanding the formula…
Mirko
- 215
1
vote
1 answer
Conditioning on a Continuous Variable with a Discrete Value
Textbook Question:
\begin{equation}
f(x,y) =
\begin{cases}
x + y & \text{if $0\le x \le1$, $0\le y \le1$ } \\
0 & \text{otherwise}
\end{cases}
\end{equation}
What is $P(X < \frac{1}{4} \mid Y = \frac{1}{3})$?
[edit: added solution for…
seven
- 11
1
vote
0 answers
Continuous conditional probability on fixed given value
I am taking a beginner's probability theory course. One thing that is confusing me a lot. We know that the probability of a continuous random variable at a fixed point is $0$. But when learning about conditional density functions in joint…
Ali Ahmed
- 11
1
vote
2 answers
True or false: For continuous random variable $X$, if $E(X)$ has a closed form, then $P(X
For this question, let us define "closed form" as an expression restricted to addition, subtraction, multiplication, and division; exponents and logarithms, including $e^x$ and $\ln{x}$; trigonometric functions and inverse trigonometric functions.…
Dan
- 35,053
1
vote
0 answers
What's the exact meaning of maps that "continuously depends on some parameters"?
In many cases we can see statements like "view $\theta$ in the function $f(\theta, x)$ as parameters and $f$ as function $f_{\theta}$ on $x$". Intuitively we guess if $f$ is continuous the $f_{\theta}$ "continuously depends on $\theta$".
More…
Albert Liu
- 61
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