Intereting Posts

Every path has a simple “subpath”
On the smooth structure of $\mathbb{R}P^n$ in Milnor's book on characteristic classes.
Dulac's criterion and global stability connection
Tips on writing a History of Mathematics Essay
Interpretation of $\epsilon$-$\delta$ limit definition
Could you recommend some classic textbooks on ordinary/partial differential equation?
Why don't we allow the canonical Gaussian distribution in infinite dimensional Hilbert space?
Calculate $\mathbb{E}(W_t^k)$ for a Brownian motion $(W_t)_{t \geq0}$ using Itô's Lemma
Fallacy limit problem – Where is the mistake?
How to prove that nth differences of a sequence of nth powers would be a sequence of n!
Assume that $(\text{X}, T)$ is compact and Hausdorff. Prove that a comparable but different topological space $(\text{X},T')$ is not.
What is an intuition behind total differential in two variables function?
Prove $\ker {T^k} \cap {\mathop{\rm Im}\nolimits} {T^k} = \{ 0\}$
General Lebesgue Dominated Convergence Theorem
Is there a simple proof for ${\small 2}\frac{n}{3}$ is not an integer when $\frac{n}{3}$ is not an integer?

Denote the pdf of the standard normal distribution as $\phi(x)$ and cdf as $\Phi(x)$. Does anyone know how to calculate $\int_{-\infty}^y \phi(x)\Phi(\frac{x−b}{a})dx$?

Notice that this question is similar to an existing one,

https://mathoverflow.net/questions/101469/integration-of-the-product-of-pdf-cdf-of-normal-distribution

- Application of Picard-Lindelöf to determine uniqueness of a solution to an IVP
- Compute: $\sum\limits_{n=1}^{\infty}\frac {1\cdot 3\cdots (2n-1)} {2\cdot 4\cdots (2n)\cdot (2n+1)}$
- This integral is defined ? $\displaystyle\int_0^0\frac 1x\:dx$
- Evaluate limit of integration $\lim_{x \to \infty} (\int_0^x e^{t^2} dt)^2/\int_0^x e^{2t^2} dt$
- Why does my professor say that writing $\int \frac 1x \mathrm{d}x = \ln|x| + C$ is wrong?
- Least sum of distances

the only difference being that I’m computing the integral over $(-\infty, y)$ for some real $y$, rather than over the entire real line.

Thank you!

- Let $f(x)$ be differentiable at $\mathbb R$, s.t $|f^\prime (x)| \le 6$ . Its given that $f(1)=20$, and $f(9)=68$ , prove that $f(7)=56$.
- Evaluating this integral : $ \int \frac {1-7\cos^2x} {\sin^7x \cos^2x} dx $
- A limit question (JEE $2014$)
- Integral $\int_1^\infty\frac{\operatorname{arccot}\left(1+\frac{2\pi}{\operatorname{arcoth}x-\operatorname{arccsc}x}\right)}{\sqrt{x^2-1}}\mathrm dx$
- How to prove that $\lim_{n \to\infty} \frac{(2n-1)!!}{(2n)!!}=0$
- Solving $\lim_{n\to\infty}(n\int_0^{\pi/4}(\tan x)^ndx)$?
- integral of $\ln x$ from 0 to 1
- inverse function and maclaurin series coefficients.
- Evaluate $\displaystyle I=\int _{ 0 }^{ 1 }{ \ln\bigg(\frac { 1+x }{ 1-x } \bigg)\frac { dx }{ x\sqrt { 1-{ x }^{ 2 } } } }$
- Integral $\int_{0}^{\infty}e^{-ax}\cos (bx)\operatorname d\!x$

As already explained, when $a\gt0$ the full integral is $1-\Phi\left(b/\sqrt{a^2+1}\right)$. The same approach shows that the integral considered here is

$$

I(y)=P(Y\leqslant(X-b)/a,X\leqslant y),

$$

where $(X,Y)$ are i.i.d. standard normal, that is,

$$

I(y)=P(aY+b\leqslant X\leqslant y).

$$

I see no reason to expect more explicit formulas.

If you check out the integral tables in section 4.2 and 4.3 of http://nvlpubs.nist.gov/nistpubs/jres/73B/jresv73Bn1p1_A1b.pdf, you will find what you need to get this done. I used equations 4.3.2 and 4.2.1 (latter is same as eqn 7.4.36 in Abranovitz and Stegun). Just change variables on the error function and complete the square on the exponential. You will end up with one 4.3.2 integral, one 4.2.1, and one simple erf(x) integral of the exponential square.

I am a bioinformatician. This problem occured for me in the context of statistics. I was trying to compute conditional probabilities to input in my factor graph model.

I verified equation 4.2.1 from the source litterature. I will have to doublecheck if 4.3.2 is correct (via numerical integration), since this solution is original to this work.

$$\int_{-\infty}^y \phi(x) \Phi(\frac{x-b}{a})dx = BvN\left[\frac{-b}{\sqrt{a^2+1}}, y; \rho= \frac{-1}{\sqrt{a^2+1}}\right]$$

where $BvN(w, z; \rho)$ is the bivariate normal cumulative with upper bounds $w$ and $z$, and correlation $\rho$.

For reference, see equation (10,010.1) in Owen (Comm. in Stat., 1980).

It’s quite easy to calculate the indefinite integral by integrating by parts, using $\Phi (x)^{‘} = \phi (x)$. The result is then straightforward.

- $R$ with an upper bound for degrees of irreducibles in $R$
- How to resolve this absolute value inequality $|1+x^2|>|x|$?
- Finding the closest point in a set to another point in n-dimensional space: efficiently
- Infinite product
- Roots of a polynomial in an integral domain
- Proving that $S=\{\frac{1}{n}:n\in\mathbb{Z}\}\cup\{0\}$ is compact using the open cover definition
- How many arrangements of a (generalized) deck of (generalised) cards have pairs in them?
- For all infinite cardinals $\kappa, \ (\kappa \times \kappa, <_{cw}) \cong (\kappa, \in).$
- Prove Euler's Theorem when the integers are not relatively prime
- Are there any memorization techniques that exist for math students?
- Solving the heat equation with Fourier Transformations
- Expectation of Minimum of $n$ i.i.d. uniform random variables.
- Why there is much interest in the study of $\operatorname{Gal}\left(\overline{\mathbb Q}/\mathbb Q\right)$?
- Hom-functor preserves pullbacks
- Show that the only tempered distributions which are harmonic are the the harmonic polynomials