1
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1answer
37 views

Differentiating Integrals

This problem appears as example 2d of Chapter 5 in "A First Course in Probability - Ross, 8th ed." Suppose that if you are s minutes early for an appointment, then you incur the cost cs, and if you ...
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1answer
18 views

Deriving marginal effects in multinomial logit model

For the multinomial logit model, it holds that: $$P[y_i=j]=\frac{\exp{\beta_{0,j} + \beta_1 x_{ij}}}{\sum_h \exp(\beta_{0,h} + \beta_1 x_{ih})}$$. Now my book states that the marginal effect is as ...
1
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0answers
31 views

Gateaux derivative, expected value

Suppose F,G be the distribution functions and \begin{eqnarray*} T(F) & = & \int xdF=E_{F}(x),\\ T(G) & = & \int xdG=E_{G}(x),\\ F_{t} & = & (1-t)F+tG, \end{eqnarray*} How to ...
2
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1answer
204 views

Derivative of double summation and dot notation?

I am trying to differentiate the following summation: $$ L(\mu, \tau_1, \ldots, \tau_i)= \sum_{i=1}^v \sum_{t=1}^{r_i} (y_{it}-\mu - \tau_i)^2 $$ $$\frac{dL}{d\mu} = y_{\cdot\cdot}-n\mu - ...
1
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1answer
64 views

Why does this interchanging of derivative and sum work?

I'm reading a stats book and, for a geometric distrubution ($E[Y]=p \sum_{y=1}^{\infty}yq^{y-1})$ it makes the claim that since $\displaystyle \frac{d}{dq}(q^y)=yq^{y-1}$ hence $\displaystyle ...
5
votes
2answers
115 views

Functions whose derivatives can be written as a function of themself?

What kinds of function $f: \mathbb{R} \to \mathbb{R}$ can be written as some function of itself? I.e. $f'(x) = g(f(x))$ for some function $g$? If $f$ is given, can $g$ be solved in terms of the ...
0
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1answer
27 views

Stuck on 'differentiating the integral from above' for computing of a PDF

I am stuck on a math derivation that has to do with statistics, so I am putting the statistical context here for context. In short, I am stuck on understanding how the answer to the PDF was attained. ...
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2answers
84 views

What is the trick in the derivation? Density of a complicated function

Through one of the proofs I found a problem that really cannot solve. Imagine some density function f(x). Now, imagine that the argument is a function of the form x+c(f'(x)/f(x)). Therefore, the ...
2
votes
1answer
2k views

Taking the derivative of definite integral?

I'm having trouble understanding the derivative of definite integral. For example, why is the following true? $\frac{d}{dx}\displaystyle\int_{0}^{x}F_{1}(x-v)f_{1}(v)\, \mathrm{d}v = ...
2
votes
1answer
133 views

Stein's Paradox and James Stein Estimator

I am trying to figure the intermediate steps in the proof of the Stein's paradox. How does it go from the left to the right? $$\frac{\partial}{\partial Y_i} \left\{\frac{Y_i }{\sum_j Y_j^2}\right\} = ...
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0answers
86 views

Numerical derivative without knowing change in variable

Suppose that we have a multivariate function $f(a,b,c,a_1,b_1)$, where $a,b,c,a_1,b_1$ are all real numbers. $f = \frac{{c({a^2} + {b^2}) + {{({a^2} + {b^2})}^{1/2}}}}{{c{{({a^2} + ...
1
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1answer
101 views

Trying to understand LMS algorithm

This is what's written: So, $h(x)$ is basically a linear function to predict values of some training set. I understand everything that's happening up to the point where we take the partial ...
1
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1answer
56 views

How to derive the equations in 3:19-3:30 provide in a MIT opencourse ware lecture about the least square method?

How to derive the equations in 3:19-3:30 provide in a MIT opencourse ware lecture about the least square method? Link: http://www.youtube.com/watch?v=YwZYSTQs-Hk Thanks in advance!
3
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2answers
186 views

Differentiating the posterior distribution function

I am learning about Bayesian statistics and I'm currently doing loss functions. Let $f(\theta | \mathbf{x} ) $ be a posterior pdf . Let $F(\theta | \mathbf{x} ) $ be the associated distribution ...
0
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2answers
111 views

How to take derivative of this likelihood function?

I am working with the probability likelihood function $$ \log \prod\limits_{i=1}^{n} x_i^{y_i} + \log \prod\limits_{i=1}^{n}\left(1-{{x}_{i}}\right)^{n_i-y_i}. $$ I want to take the derivative with ...
2
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2answers
175 views

Statistics: Minimized parameters of an error function in regression

My question today is about the minimization of an error function with two parameters. It is a function that measures the error of a set of points. The two parameters are the weights of a regressor. ...