OK I've done this and I get the inequality I need. But can I just check, I don't understand how I have used the equation I need to in the OP?
For the second part if I know that -x<= sinx <= x then -1<=nx(sin(1/nx)) <= 1 but then I'm a bit stuck
Homework Statement
I am attempting to show that -x \leq sin(x) \leq x for x>0 and thus \int^1_0 nxsin(\frac{1}{nx})dx converges to 1.
Homework Equations
I know that I need to use the fact that I have shown that the limit as T tends to infinity of \int^T_1 \frac{cos(x)}{\sqrt{x}}dx...
Homework Statement
Show from definition that if f is measurable on [a,b], with m<=f(x)<=M for all x then its lebesgue integral, I, satisfies
m(b-a)<=I<=M(b-a)
Homework Equations
The Attempt at a Solution
I know that the definition is that f:[a,b]->R is measurable if for each t...
I can't believe it!
So can I just check I have understood it. So to find the integral I need to say how the limits change depending on whether x>0 or x<0 and then calculate the integral for each case?
1. so the integral would be between -1 and x+1 of 1dy which would be (x+1)-(-1)=x+2?
2.If x>0 then the integral is between 1 and x-1 of 1dy which would be (x-1)-(1)=x-2
1. This is the bit I'm struggling with i think?
2. I think so, is it because when x < 0 the overlap is given by 2+x and when x<0 the overlap is 2-x?
3. Because then x-y is not in (-1,1) and so the indicator function is zero and so the integral is zero
Erm to be honest I just used the previous question :uhh:, I thought that was all I needed to do?
Do you mean that is it because the overlap is like a triangle function?
I'm not sure but is the measure of the overlap I worked out not the answer to the integral?
So the integral will be equal to 2-|x| i.e 2+x when x is negative and 2-x when x is positive?
I think I understand what the overlap is. Is the overlap not the values the make the integral non-zero.
So to do the integral instead of the limits being 1 and -1 they will become -1 and ?
I think I have been a bit stupid here, there is a question before this that I think answers what you are asking. I had to measure the overlap of the intervals (-1,1) and (-1+x,1+x) for various values of x. I found that for values of x not in (-2,2) the overlap is empty and equals zero. For...
Sorry I think I am getting confused. I have done the graph like you said, so y must be between -1 and 1. So if x=-1.5 then I(x-y) = 1 for y between -0.5 and -1?
I understand what you mean, so I have y in (-1,1) and x in (2,-2) but I don't see a formula between the two that will make sure x-y is always in (-1,1)?
Homework Statement
Calculate f*f where f is the indicator function (-1,1)
Homework Equations
The convolution f*g of functions f and g is defined by:
f*g(x)=\int^{\infty}_{-\infty} f(x-y)g(y)\ dy
The Attempt at a Solution
I haven't really done convolution before as I am teaching myself, so...
Homework Statement
calculate the deviance for the poisson model and show that the i'th component of the poisson deviance is non-negative
Homework Equations
Dev = 2[L(y)-L(m)] where L is the log-likelihood and m is the mle
The Attempt at a Solution
I have calculated the log...
Homework Statement
Define a sequence (fn) from n=1 to infinity of functions on [0,1] by fn(t)=t^n
does the sequence converge in (CL^2[0,1],||.||2)
Homework Equations
The Attempt at a Solution
I am struggling on where to start. I am fairly new to the L2 space and so would just...
Homework Statement
Prove that a point,a, not belonging to the closed set B has a non-zero distance from B. I.e that dist(a,B)=inf(y in a) ||a-y||>0
Homework Equations
I have no idea how to start this. It is only worth a few marks and I have been told it is fairly easy but I have always...
Hey guys,
I have a function y=(x+2)/(x+1) and I have performed iterations to show that for any initial value other than -sqrt{2} the sequence converges to sqrt{2}.
So I have found that sqrt{2} is a stable fixed point and -sqrt{2} is an unstable fixed point.
Now I have to prove my...