For question involving exponential functions and questions on exponential growth or decay.

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2
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1answer
33 views

Distribution of $Z$ from Moment Generating Function

Suppose that $X_1, X_2, ..., X_n$ are independent and identically distributed Exp(λ) random variables and let $Z = X_1 + X_2 + · · · + X_n$. Determine $M_Z(θ)$, the moment generating function of $Z$ ...
1
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2answers
37 views

Holomorphic Functional Calculus for Exponential Identity

For a Banach algebra $A$ with unit $e$, then for $a\in A$, I want to prove $$\exp((\alpha+\beta)a)=\exp(\alpha a)\exp(\beta a)\qquad\text{for all }\alpha,\beta\in\mathbb{C}$$ So far I have said let ...
2
votes
1answer
29 views

Moment Generating function hard example!

X is a random variable with density $$f(x)=2e^{-2x+2} , x\geq1$$ and 0 otherwise. Determine $Mx(θ)$, the moment generating function for X, and the values of θ for which $Mx(θ)$ is defined. Use ...
2
votes
2answers
34 views

exponential functions with constant

I'm in a pre-calc class, and we're looking at logarithms and exponential functions. One of the exercises I'm struggling with is: $$5e^{2x} = 6 + 29e^x$$ I would ususally multiply each side by a log ...
1
vote
1answer
37 views

Convergence/Divergence of Series for $e^{-1}$

How do I show that the series $\sum \frac{n^n}{n!} x^n$ diverges at $x =e^{-1}$? I understand that $a(n) = (\frac{n}{n+1})^n$ is a strictly decreasing function and therefore $(\frac{n}{n+1})^n > ...
0
votes
0answers
35 views

Expected values with e to the power of e^x as a factor

Suppose that the random variable X has an exponential distribute with mean four. Let the random variable Y=ln(X+1). Find E[Y] if $y=ln(x+1)$ then it follows that $x=e^y-1$ and that $dx/dy = e^y$ ...
2
votes
1answer
85 views

Show that $e^x=1+x+\frac{x^2}{2!}+…+\frac{x^n}{n!}+R_{n+1}$

Show that $\qquad$ $\qquad$ $e^x=1+x+\frac{x^2}{2!}+...+\frac{x^n}{n!}+R_{n+1}$ with $\qquad \qquad$ $0 \lt R_{n+1} \lt e^x \frac{x^{n+1}}{(n+1)!}$ if $0 \lt x$ and $\qquad \qquad$ $|R_{n+1}| \lt ...
0
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1answer
14 views

Writing a doubling equation given only amount of time to double.

Write an equation to model the generation of Ecoli if the doubling time is 20 minutes. I can't for the life of me figure this out.
1
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2answers
51 views

Integrating exponential function raised to a fractional term

I'm trying to integrate an exponential term raised to a fractional power with other variables in it. I'm really rusty and having a hard time trying to figure out where to start. I'd like to pull out ...
0
votes
1answer
24 views

Memory-less property of Exponential distribution.

X follows exponential distribution, when $p.d.f.$ of random variable $X$ is $$f(x)=\lambda e^{-\lambda x},\ x \ge 0$$ , where $E(X)=\displaystyle\frac1\lambda$. Problem A cellphone, which can be ...
1
vote
1answer
35 views

Density function of uniform prob distribution

Let $X ∼\operatorname{Uniform}(0,1)$. Find the density function of $Y = e^X$. I got to: $F_Y(y)$=$P(Y\le y)$=$P(e^X\le y)$=$P(X\le \ln(y))$ Not sure where to go from here?
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2answers
45 views

confidence interval for median of an exponential distribution

I am having a hard time finding the confidence interval of the median of an exponential distribution. I am currently studying for an upcoming test. I found the mle of $\lambda = \frac{5}{61}$ I ...
2
votes
2answers
20 views

Conversion from exponential to cosine

I'm trying to understand the following expansion. The question was Show that if $Y(t) = X(t+a) - X(t-a)$ and $X(t)$ is WSS, then $$S_Y(\omega) = 4S_X(\omega)sin^2a\omega$$ The solution is ...
2
votes
0answers
41 views

What is the value of $\int_0^{\infty} \frac{e^{-x} dx}{1 + \ln(1+x^2)} ?$ [closed]

Im not sure how to do this integral $$\int_0^{\infty} \frac{e^{-x} dx}{1 + \ln(1+x^2)}$$
0
votes
1answer
37 views

Transformation of Exponential Random Variables

I came across this example, and I was completly thrown off! Hopefully someone will be able to help me start it or even explain it all to me! Thanks for any help! X and Y are two independent random ...
0
votes
1answer
27 views

Hyperbolic sine of a logarithm

Re-express $11\sinh(\ln 8)$ in the form $n/m$ where n and m are integers. I am not sure where to start. Never went over something like this, its probably very easy though.
0
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4answers
67 views

Derivative of $4x^{5x}$

I'm studying for an exam and I'm confused on these type of derivative problems. I know the answer I'm just confused as how to get to the answer. Would anyone mind going through the steps: Question: ...
0
votes
0answers
25 views

For $X$ exponential with mean $\frac{1}{\lambda}$, find pdf of $X^2, X^3,$ and $e^{-\lambda X}$

Supposed to express the answers in terms of $\lambda$. I tried X^2 and did $F_Y(x)=P[Y \leqslant X]=P[X^2 \leqslant x]=P[-\sqrt{x} \leqslant \sqrt{x}]$ Then this equals $F_X(\sqrt{x})-F_X(-\sqrt{x})$ ...
0
votes
0answers
75 views

How do I show that $y(t)=\int_0^tv(t-s)f(s) ds$ is a solution of $L[y]=f(t)$?

We have the $n$th-order scalar differential equation $$L[y]=\frac{d^{n}y}{dt^{n}}+a_1\frac{d^{n-1}y}{dt^{n-1}}+\dots+a_ny=f(t).$$ Let $v(t)$ be the solution of $L[y]=0$ which satisfies the initial ...
0
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1answer
30 views

Explanation for using algebra to solve problems

Here is the question: Thirty minutes after a patient is administered his first dose of a medication, the amount of medication in his blood stream reaches 224 mg. The amount o medication in ...
0
votes
1answer
32 views

Distribution of Arriving students?

I am trying to solve the following question: A class has 20 students, after the first month, the professor has figured out that arrival times T1, T2, ... ,T20 of students are independent and ...
0
votes
1answer
24 views

Exponential $2\pi$ in DTFT

While studying Discrete-time Fourier Transform. I found that $X[\omega]$ is periodic with $2\pi$. So when did the proof: $$ X[\omega + 2\pi] = \sum_ {n= +\infty}^{-\infty}x[n]e^{-j \omega n}e^{-j 2\pi ...
1
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0answers
31 views

Compute $\int_0^1 \frac{\exp(u^3 - 4/3 u) du }{\sqrt {1 - u^2}} $

Compute the integral $$\int_0^1 \frac{\exp(u^3 - 4/3 u) du }{\sqrt {1 - u^2}} $$ I assume contour integration is the easier way ?
1
vote
1answer
12 views

Determining the distribution of univariate transformation

If Y is uniformly distributed on the interval $(0, 1)$ and if $Z = –a * ln(1 – Y)$ for some $a > 0$, then to which of the following families of distributions does Z belong? Lognormal ...
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6answers
52 views

how to find an infinity limit in a fraction

I don't understand how to find the limits of this expression when $x\to\infty$ and $x\to-\infty$: $$\left(\frac{3e^{2x}+8e^x-3}{1+e^x}\right)$$ I've searched for hours. How to compute these limits?
0
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1answer
43 views

How does wolfram alpha compute this sum of exponentials?

I have been breaking my head from morning, but I have not been able to understand how wolfram alpha obtained the relationship. Link
-1
votes
2answers
44 views

Proving an inequality involving an exponential and a polynomial? [closed]

$$106+2^{x}>2x^{2}+2x$$ I have tried to prove that this inequality is true for $x\ge 0$. I would be happy for any hint! Thanks for the two answers! They are very helpful. But I am interested in a ...
1
vote
1answer
40 views

Why does $u=e^{-i\omega x}e^{-k\omega^2t}$ “clearly” solve the 1-D heat equation?

So one of my least favorite things that textbooks do is using the words "clearly", "it should be obvious", etc. In my PDEs class, we've started the Fourier Transform, and I missed the first day of it ...
1
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1answer
41 views

Help with integration involving exponential

I am trying to solve an equation in the book Digital Image Processing, but I am stuck in the steps in between the formula and solution. Here's the equation, the last line is the solution. Sorry it's ...
2
votes
5answers
107 views

Solve equations like $3^x+4^x=7^x$

How can I solve something like this? $$3^x+4^x=7^x$$ I know that $x=1$, but I don't know how to find it. Thank you!
0
votes
0answers
17 views

Fundamental matrix with nilpotent power

Given that a square $n\times n$ matrix $\bf{A}$ has only one eigenvalue $r$ with multiplicity $n$, we have by Cayley-Hamilton Theorem that $(\bf{A}$$-r$$\bf{I})^n$=$\bf{0}$. We can then express ...
1
vote
1answer
41 views

How to prove these two functions are equal on the interval [-1,0]

I created a function by first considering some well known limits: $$\lim_{n\to\infty } \frac{n}{2}\sin\left ( \frac{2\pi}{n} \right )=\pi $$ $$\lim_{n\to\infty } \left ( 1+\frac{1}{n} \right ...
0
votes
2answers
30 views

$\exists C>0$ such that $\frac{1}{2}(e^x+e^{-x}) \le e^{\frac{1}{2}x^2-Cx}$?

It is well-known and easy to check that for any real $x$ it holds $$ \frac{1}{2}(e^x+e^{-x}) \le e^{\frac{1}{2}x^2}. $$ [To show this, it is sufficient to write explicitly their Taylor series ...
1
vote
2answers
53 views

Is it possible to inverse a sum of exponents

I have a problem, I need to inverse a sum of exponents. Is it possible? I have this function $y = 0.84826731\times e^{-1.10973369x} + 0.17939312\times e^{-0.1902204x} + 0.02965983\times ...
3
votes
2answers
71 views

Does $ \int \frac{exp( -b(a+x)^{3/2})}{\sqrt{x}} dx$ have a solution?

Is there a solution for the following integral: $$ \int\frac{\exp(-b(a+x)^{3/2})}{\sqrt{x}} dx $$ where $a$ and $b$ are constants. If it is not, what is the best approximation? Especially in the ...
5
votes
1answer
95 views

Integral of $\int x^{-x} dx$

Question: $\int x^{-x} dx =$ ? Hint: $$ e^{x\ln \frac{1}{x}} = \sum_{n=0}^\infty \frac{x^n}{n!} \left(\ln\left(\frac{1}{x}\right)\right)^n$$ I figure since $\int x^{-x} dx = \int e^{x\ln ...
4
votes
3answers
55 views

$\lim_{x\to 0+}=(1/x)^{\sin x}$? [duplicate]

$\lim_{x\to 0+}=(1/x)^{\sin x}$ I think I should rewrite it into a from $e^{\ln}$ , but I can't continue the calculation after this step.
5
votes
4answers
69 views

Limit of $\frac{(n+1)^{2n}}{(n^2+1)^n}$ as $n\to \infty$

So it is given to find $$\lim_{n\to \infty}\dfrac{(n+1)^{2n}}{(n^2+1)^n}$$ So what I did is $$\lim_{n\to \infty}\dfrac{(n+1)^{2n}}{(n^2+1)^n}=\lim_{n\to ...
3
votes
1answer
62 views

What is the $\int_{-1}^{1} (4t^4 - 4t^2 - 1) e^{-{(t^2 -1)}^2}dt$?

I tried doing this by substitution $-((t^2 -1)^2) = u$ So $-4t(t^2 -1)dt = du$ but I don't know what to do with the "-1" in $(4t^4 - 4t^2 -1) = (4t^2(t^2 -1) -1)$
3
votes
3answers
137 views

What is the inverse function of $e^x +x$?

As the natural $\log(x)$ function is the inverse of the exponential $e^x$ and $\log(x +1)$ is the inverse of $e^x - 1$, what it the inverse of $e^x + x$?
1
vote
1answer
31 views

Two exponential terms equation solution

Let $A_i$ and $B_i$ denote constants, I know this equation $$A_1 \exp(B_1x) + A_2x + 1 = 0$$ can be solved using lambert W function. But can I get a general solution of this equation? $$A_1 ...
0
votes
1answer
48 views

How to simplify $\ln{\left(x + \ln{\left(x + \ln{\left(x + …\right)}\right)}\right)}$.

I have tried the following: $s = \ln{\left(x + \ln{\left(x + \ln{\left(x + ...\right)}\right)}\right)}$ $s = \ln{\left(s + x\right)}$ $e^{s} = s + x$ However, I am unsure as to how to proceed. ...
1
vote
2answers
42 views

Approximate an exponential factor

What math methods can I use to approximate lambda in the following system of equations?: $$ e^{-0.05\lambda}=0.5469\\ e^{-0.1 \lambda} = 0.3229\\ ...\\ e^{-0.2 \lambda} = 0.1226$$ I am trying to fit ...
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vote
2answers
46 views

Properties of Exponential Matrix [duplicate]

One of the properties is that $e^{{\bf A}+{\bf B}}\neq e^{\bf A}e^{\bf B}$ unless ${\bf AB}$$={\bf BA}$. Can someone please explain how exactly commutativity matters in this case? I'm guessing it has ...
0
votes
1answer
56 views

How to find the solution to this summation

This was a question asked in our exam and we have to write a code for it. We have to find the summation of following series $\log(\sum_1^n (e^{x_i}))$ where $1 < n < 10^6$ and $0 < x_i < ...
3
votes
3answers
58 views

Is there a whole number $x\in\mathbb{Z}$ with $x\neq 0$ s.t. $\exp(x)$ is natural?

I was wondering if there is a number $x\in\mathbb Z$ with $x\neq 0$ s.t. $\exp(x)\in\mathbb N$ and if not, why is that so? EDIT: Forgot to exclude the $0$.
0
votes
1answer
103 views

Find the area of the biggest rectangle that can be inscribed under the graph?

Find the area A of the largest rectangle that can be inscribed under the curve of the equation below in the first and second quadrants. $$y = e^{-x^2}$$ Graph of the equation. I don't know where ...
1
vote
1answer
30 views

Solve the following inequalities

Does there exist a general formula for comparing two prime numbers powers . Ex. $$ max(2^{2002},3^{1335} )$$ or $$ max(2^{2004} ,3^{1202} ) $$ NOTE: Im a twelfth grader and i accidentally stumbled ...
0
votes
2answers
63 views

Integration of $e^{-x}$ with respect to y

I'm not sure if I'm being incredibly stupid and having a brain dead moment! any help is appreciated! The question I'm referring to is dealing with the integration of an exponential function of x with ...
21
votes
6answers
2k views

Which of the following powers is bigger: $2^{41}$ or $3^{24}$?

Calculate the maximum between the given numbers $$ \max(2^{41},3^{24})\text{.}$$ I got stuck when I tried to decompose the exponent as $41$ is a prime number.