Exponential Definition and 1000 Threads

[SIZE="4"]Definition/Summary The exponential distribution is a probability distribution that describes a machine that it equally likely to fail at any given time. [SIZE="4"]Equations f(t) = e^{-\lambda t} \lambda [SIZE="4"]Extended explanation A machine is equally likely to fail...

[SIZE="4"]Definition/Summary The exponential (the exponential function), written either e^x or exp(x), is the only function whose derivative (apart from a constant factor) is itself. It may be defined over the real numbers, over the complex numbers, or over more complicated algebras such as...

Homework Statement Homework Equations — The Attempt at a Solution Confused with (d) a little. Rocket explodes at ##h=3.85262 ## miles ## -16t^2+1400\sin(\alpha)t=3.852624*5280## ## \alpha=\arcsin\left(\dfrac{3.852624*5280+16t^2}{1400t}\right) ## But what do I do...

Homework Statement Dear all I am calibrating a temperature measurement model and I am stuck with an equation. The variable z is given; x and y represent two regression terms with common regressors - which I will solve for a specific regressor in a second step. Homework Equations...

[I don't know if this is in the right topic or not so I hope I'm all good] My question is related to the exponential growth and decay formula Q=Ae^(kt). Simply, why is the value e used as the base for the exponent? Does it have to be e? If so, can anybody tell me why? Thanks~| FilupSmith |~

For proving the natural number, e. (1+1/n)^n As n approches infinite, (1+1/n)^n ----> e However, wouldn't it become one as n becomes infinite? (1+1/n)^n=(1+0)^n=1 Could anyone explain this to me please!

Hi, I've been working on trying to model sales for my work and I'm really just modeling it using exponential regression, so I get the linear regression of the logarithm of the data and obtain the desired formula. What I'm confused about is that if I integrate this to try to predict how many...

Help? Suppose the random variable Y has an EXP(2) distribution. What is P(Y > 1)? (Round to four decimal places as appropriate.)

I'm participating in research this summer and it's has to do with the Fourier Series. My professor wanted to give me practice problems before I actually started on the research. He gave me a square wave and I solved that one without many problems, but this triangle wave is another story. I've...

Can the integral ∫0texp(a/(b+ct'))dt' be carried out analytically? Or approximated by taking the taylor expansion of exp(a/(b+ct'))

I'm going through the Degroot book on probability and statistics for the Nth time and I always have trouble 'getting it'. I guess I would feel much better if I understood how the various distribution arose to begin with rather than being presented with them in all there dryness without context...

Hi, The two terms should vanish at infinity according to the Quantum textbook of Griffiths, but I don't see how? I mean a complex exponential is a periodic function so how can it vanish at infinity? If you split up the first term exp(ikx) * exp(-ax) Take the limit of infinity...

Hi. I notice that some values of X on the exponential distribution PDF have a value of around 1. I understand the integral ends up being one, since those values of X are less than 1. But P(X) at those points still gets to 1, or thereabouts. How does that make sense, that the probability of a...

What is the significance of the standard deviation (equal to the mean) in an exponential distribution? For example, as compared to the standard deviation in the normal distribution, which conforms to the '68-95-99.7' rule? thanks

Hello! :cool: I am looking at the exponential power series: $$\sum_{n=0}^{\infty} \frac{x^n}{n!}$$ It is $R=\displaystyle{\frac{1}{\lim_{n \to \infty} \sup \sqrt[n]{|a_n|}}}=\frac{1}{\lim_{n \to \infty} \sup \sqrt[n]{n!}}=+\infty$ So,the power series converges at $(-\infty,+\infty)$,so...

ok, so I have a list of students with GPA, I checked the probability plot and I think its a Exponential distribution, take a look: So I am given a χ-bar to prove, and I have to prove or test it with three different types of test, I don't know which ones or how to do them in miniTab...

I am doing some analysis and I have come up with the following integral: \int \frac{e^{-ax}}{1+be^{-cx}}dx where a>0, b>0 and c>0. I have found out this integral has a solution in terms of the Gaussian hypergeometric function http://en.wikipedia.org/wiki/Hypergeometric_function but it...

If is possible to expess periodic functions as a serie of sinusoids, so is possible to express periodic functions with exponential variation through of a serie of sinusoids multiplied by a serie of exponentials? Also, somebody already thought in the ideia of express any function how a serie of...

I'm doing some optional problems in preparation for my final in two weeks in one of my classes and I'm stumped on this one in particular Express irrational solutions in exact form and as a decimal rounded to three decimal places. Problem: 0.3(4^0.2x) = 0.2 I won't be able to look back here...

The population (P) of an island y years after colonisation is given by the function P = 250/(1+4e^-0.01y) A. What was the initial population of the island? B. How long did it take before the island had a population of 150? C. After how many years was the population growing the fastest?

If \hat{U}(r) = e^{\hat{A}(r)}, can we say \frac{d\hat{U}}{dr} = \frac{d\hat{A}}{dr}e^{\hat{A}(r)}?

Homework Statement Let X1, X2,...,Xn be a random sample from the exponential distribution with mean θ and \overline{X} = \sum^{n}_{i = 1}X_i Show that \overline{X} ~ Gamma(n, \frac{n}{θ}) Homework Equations θ = \frac{1}{λ} MGF Exponential Distribution = \frac{λ}{λ - t} MGF Gamma...

X,Y r.v statistically independent ,with exponential Distribution. calculate the density function of X/Y (Let $X$ have distribution ${\lambda}e^{-{\lambda}x}$ and $Y$ have distribution ${\lambda}e^{-{\lambda}y}$ i know i should use transformtion u=X+Y ;v=X/Y to solve it)

Hi, I'm new to this forum so thank you in advance for any help on this problem. I would like to understand the steps needed to solve this problem. The answer is 3.94111..286 = (1.00178^(12 * t))/t or .286 = (e^.021341 * t)/t

Homework Statement Suppose f is differentiable on \mathbb R and \alpha is a real number. Let G(x) = [f(x)]^a Find the expression for G'(x) Homework Equations I'm pretty sure that I got this one right, but I really want to double check and make sure. The Attempt at a Solution...

Is there an expression, in general, for the product of two matrix exponentials, for non-commuting matrices? i.e. something of the form, e^Ae^B = e^{( * )} where the ( * ) would, I assume, depend in some way on the commutator [A,B] ? I can only find examples online when [A,B] = 0...

Hi All, I am struggling to prove the following identity $$ 1 + y + \frac{1}{2!}y^2 + \frac{1}{3!}y^3 + \dots = lim_{N \to \infty} \sum_{r=0}^{N} \frac {N!}{r! (N-r)!} \frac{y}{N}^{r} = lim_{N \to \infty} (1 + \frac{y}{N})^N $$ any hint would the most appreciated. I understand the...

My confusion comes from basic exponent rules and whether or not both sides of an equation have to have the same level of exponent, when you reduce the base for solving. If one side can have an exponent of 3, does the other side also have to be reduced to something that would also have an...

I'm having some problem in determining the phase of an exponential Fourier series. I know how to determine the coefficient which in turn gives me the series after I multiply by e^-(jωt) I can determine the amplitude by dividing the coefficient by 2 |Dn| = Cn/2 Now my question is how to...

NOTE: This isn't homework. So I'm trying to integrate a really awkward integral with limits from a to infinity; \int^{∞}_{30471.2729807}(\frac{83.1451 * 373.15}{X})-(\frac{83.1451 * 373.15}{X-30.4811353}-\frac{5534906.5380409}{X^2})dX Since the Simpson's and Trapezoidal would be really...

I'm writing a piece of software, however, my math skills are VERY rusty at the moment. The problem is as follows: There is a journey consisting of 256 steps The person, can either i) walk, or ii) hop The person can choose what to do at each step I want to compute a list of all possible...

Calculation of all real values of $x$ for which $3^x+6^x = 4^x+5^x$

Show that $ \displaystyle \text{Im} \ \exp \left( ae^{be^{i c \ x}} \right) = \exp \left( ae^{b\cos(cx)\cos (b\sin cx)} \right) \sin \left( a\sin (b\sin c x) e^{b\cos cx} \right) $.EDIT: Then by integrating $ \displaystyle f(z) = \frac{z \exp ( ae^{be^{ic \ x}} )}{z^{2}+d^{2}}$ around a...

This is part of a differential equations group project problem where I solve a set of differential equations to obtain the solution to a function. The part that I am stuck at involves taking the log of an exponential function, though there may be a mistake on the book's part, but I'm not sure...

Calculation of real values of $x$ in $\sqrt{4^x-6^x+9^x}+\sqrt{9^x-3^x+1}+\sqrt{4^x-2^x+1} = 2^x+3^x+1$ My Try:: Let $2^x = a$ and $3^x = b$ , Then $\sqrt{a^2-a\cdot b+b^2}+\sqrt{b^2-b+1}+\sqrt{a^2-a+1} = a+b+1$ Now I am struck after that Help required Thanks

Is exponential function e to the power x[t] stable? My book uses BIBO and says its stable but for -x[t] it says its not stable. Is my book wrong?

Homework Statement Hey guys. So here's the situation: Consider the Hilbert space H_{\frac{1}{2}}, which is spanned by the orthonormal kets |j,m_{j}> with j=\frac{1}{2}, m_{j}=(\frac{1}{2},-\frac{1}{2}). Let |+> = |\frac{1}{2}, \frac{1}{2}> and |->=|\frac{1}{2},-\frac{1}{2}>. Define the...

Hi MHB, Solve $(2+ \sqrt{2})^{(\sin x)^2}-(2- \sqrt{2})^{(\cos x)^2}=\left( 1+ \dfrac{1}{\sqrt{2}} \right)^{\cos 2x} -(2-\sqrt{2})^{\cos 2x}$. This problem vexes me much because the only way that I could think of to solve this problem would be by substituting $(\sin x)^2=u$, and from there, I...

Let \[ \mathbf{A} = \begin{bmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ -4 & -5 & -4 \end{bmatrix} \] Then I want to find \(e^{\mathbf{A}t}\). \[ \mathbf{I} + \mathbf{A}t +\frac{\mathbf{A}^2t^2}{2!} + \frac{\mathbf{A}^3t^3}{3!} + \cdots \] I have up to the 6th term but I can't identify the series. \[...

Show that for positive parameters $a$, $b$, and $c$, $$ \int_{0}^{\infty} \left( \frac{e^{-ax}-e^{-bx}}{x^{2}} + (a-b) \frac{e^{-cx}}{x} \ \right) \ dx = b-a + a \ln \left(\frac{a}{c} \right) - b \ln \left(\frac{b}{c} \right)$$

Homework Statement Homework Equations f(x) = e-λλx/x! The Attempt at a Solution Initially I thought I could solve this problem using the Law of Memoryless. That, the solution would just be P(X <= 2). However, I was wrong. Turns out the solution is P(X <= 4.5) - P(X<= 2.5). Does anyone know why?

If I'm given a function ##f(x) = A cos (x) + B sin (x)##, is there any way to turn this into an expression of the form ##F(x) = C e^{i(x + \phi)}##? I know how to use Euler's formula to turn this into ## \alpha e^{i(x + \phi)} + \beta e^{-i(x + \phi)}##, but is there a way to incorporate the...

Homework Statement As the title indicates. I'm given two independent exponential distributions with means of 10 and 20. I need to calculate the probability that the sum of a point from each of the distributions is greater than 30. Homework Equations X is Exp(10) Y is Exp(20) f(x) =...

Tunneling from Rectangular barrier - Exponential Decay ?? Consider the Rectangular Potential Barrier. If one solves bound state Problem in this case, wavefunctions of Exponentially Decaying and rising kind are found for the Region in the Barrier. ψ = A eαx + B e-αx Yet Most Books and...

I am told that an exponential distribution is memoryless. But why aren't other distributions, such as the normal distribution, also memoryless? If I pick a random number from an exponential distribution, it is not effected by previously chosen random numbers. But isn't that also the case for...

Has anybody got an idea how to solve this integral. I tried integration by parts, and that made the things even more complicated, substitutions as well. I used Mathematica to Solve that problem. Here is the integral...

Homework Statement Is a function that has a square root an exponential graph since you can rewrite a square root as x^1/2? Homework Equations The Attempt at a Solution

I am looking for help solving for x for the question below. Any help would be greatly appreciated. 2^x=16(8^2x)

Hi, I attached a pdf with a question and its answers. I don't understand the whole thing basically, e.g what their asking for.. 1.2.1;1.2.2;1.2.3 1.2.2 - How many solutions for x will the equation have..? 1.2.3 - Largest integer for which it will have solutions..? Any explanations will...

Doing some self prep for Diff EQ starting next week. Determine the decay rate of C14 which has a 1/2 life of 5230. Using e^kt as a function, I solve using k5230=ln.5 which gives the obvious answer of negative what I want. How do I know to use the reciprocal (ln2) other than to "just know" I...