Consider the function f(\mu)=\sum_{i=1}(x_{i}-\mu)^{2}, where x_[i}=i,\ i=1,\ 2,\ \cdots,\ n What is the

Bobbie Comstock 2021-12-16 Answered
Consider the function
\(\displaystyle{f{{\left(\mu\right)}}}=\sum_{{{i}={1}}}{\left({x}_{{{i}}}-\mu\right)}^{{{2}}},\) where
\(\displaystyle{x}_{{{i}}}={i},\ {i}={1},\ {2},\ \cdots,\ {n}\)
What is the first and second derivative os \(\displaystyle{f{{\left(\mu\right)}}}\)?

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

Janet Young
Answered 2021-12-17 Author has 1413 answers
Step 1
In the 1st step on line 1, the chain rule is applied. The term within the brackets is differentiated, producing a -1.
Then the bracket itself is differentiated, producing the 2 at the front. The 2nd step on line 1 involves no differentiation. Instead, the bracket is split into two terms.
The second term has an n because it is simply the summation from \(\displaystyle{i}={1}\) to \(\displaystyle{i}={n}\) of a constant.
The summation of a constant is equal to n multiplied by the constant. Then for the second line, there are no extra rules.
The first term becomes 0 because it's a constant and the second term loses \(\displaystyle\mu\).
Not exactly what you’re looking for?
Ask My Question
Answered 2021-12-18 Author has 3123 answers
Step 1
Adding an answer here to further clarify the other ones which are simply answers without steps. To get the first derivative, this can be re-written as:
After that it's standard fare chain rule
Second derivative: you can observe the same property of linear summation:
Answered 2021-12-28 Author has 9672 answers

Step 1
Here we have one function \((x_{i}-\mu)\) that is part of a larger function in that it is raised to a power of 2, \((\mu-i)^{2}\)
According to the extended power rule, we multiply the derivative of the outer function
the derivative of the inner function \((x_{i}-\mu)\)
The derivative of the outer function brings the 2 down in front as
and the derivative of the inner function
\((x_{i}-\mu)\) is -1.
So the -2 comes from multiplying the two derivatives according to the extend power rule:
\(f'(\mu)=-2\sum_{i=1}^{n}(x_{i}-\mu)\) and \(f''(\mu)=2n\)


Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2021-12-07
Solve this differential equation:
where y is a one-variable function x. What am I supposed to do with the cosx?
asked 2021-11-19

Consider the following linear difference equation
\(\displaystyle{f}_{{k}}={1}+{\frac{{{1}}}{{{2}}}}{f}_{{{k}+{1}}}+{\frac{{{1}}}{{{2}}}}{f}_{{{k}-{1}}},\ {1}\le{k}\le{n}-{1}\)
with \(\displaystyle{f}_{{0}}={f}_{{n}}={0}\). How do i find solution?

asked 2021-05-17

Express the limits as definite integrals. \(\displaystyle\lim_{{{P}\rightarrow{0}}}{\sum_{{{k}={1}}}^{{{n}}}}{\left({\frac{{{1}}}{{{C}_{{{k}}}}}}\right)}\triangle{x}_{{{k}}}\), where P is a partition of [1,4]

asked 2021-12-20
If we consider the equation
\(\displaystyle{\left({1}-{x}^{{2}}\right)}{\frac{{{d}^{{2}}{y}}}{{{\left.{d}{x}\right.}^{{2}}}}}-{2}{x}{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}+{2}{y}={0},\ -{1}{ < }{x}{ < }{1}\)
how can we find the explicit solution, what should be the method for solution?
asked 2021-11-23
Consider, \(\displaystyle{a}{y}{''}+{b}{y}'+{c}{y}={0}\) and \(\displaystyle{a}\ne{0}\) Which of the following statements are always true?
1. A unique solution exists satisfying the initial conditions \(\displaystyle{y}{\left({0}\right)}=\pi,\ {y}'{\left({0}\right)}=\sqrt{{\pi}}\)
2. Every solution is differentiable on the interval \(\displaystyle{\left(-\infty,\infty\right)}\)
3. If \(\displaystyle{y}_{{1}}\) and \(\displaystyle{y}_{{2}}\) are any two linearly independent solutions, then \(\displaystyle{y}={C}_{{1}}{y}_{{1}}+{C}_{{2}}{y}_{{2}}\) is a general solution of the equation.
asked 2021-11-19
We have the following differential equation
i found that the general solution of this equation is
where b and d are constats
Please how we found this general solution?
asked 2021-05-16
Let \(X_{1}....,X_{n} and Y_{1},...,Y_{m}\) be two sets of random variables. Let \(a_{i}, b_{j}\) be arbitrary constant.
Show that
\(Cov(\sum_{i=1}^{n}a_{i}X_{i},\sum_{j=1}^{m}b_{j}Y_{j})=\sum_{i=1}^{n}\sum_{j=1}^{m}a_{i}b_{j}Cov(X_{i}, Y_{j})\)