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Compute $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2+(x-y)^2+y^2)}dxdy$

Apr 2, 2017

Does $\lim\limits_{n\rightarrow \infty}E[X|F_n]$ exists a.s if $F_n$ is the $\sigma-$field generated by $Y_0,...,Y_n$ where $Y_n=X+W_n$.

Mar 10, 2017

Find the minimum value of $f$ on $\{(a,b,c)\in \mathbb{R^3}\mid a+b+c=1\}$ where $f(a,b,c)=\int_{0}^1 (a+bx+cx^2)^2dx$.

Apr 2, 2017

Show that there exists $g:D\rightarrow \mathbb{C}$ is analytic and one-to-one such that $(g(z))^2=f(z^2).$

Feb 9, 2017

For $0<a<b$ and $n>0$, show that $\left|\int_{a}^b \frac{\cos x}{x^n}\,dx\right|\leq \frac{2}{a^n}.$

Apr 5, 2017

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For integrable function $f $, $\int_{E_n}f d\mu\rightarrow 0$ where $E_n=\{x:f(x) > n\} $

Apr 23, 2017

Let $f: [a,b] \rightarrow \mathbb R $ be a Riemann integrable function. Prove that $ \lim_{c \rightarrow b-} \int_a^cf =\int_a^bf$.

Apr 22, 2017

Show that $\liminf_{n \to \infty} x_n+ \liminf_{n \to \infty} y_n \leq \liminf_{n \to \infty}(x_n + y_n)$

Mar 10, 2017

A homework problem on probability theory

Apr 18, 2017

Prove or disprove $E[(X-E[X|Y,Z])^2]\leq E[(X-E[X|Y])^2]$

Mar 11, 2017

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