1
vote
0answers
97 views

What is the solution of the integral (product of two standard normal CDFs)?

I need to compute this kind of integral: where $b>0,d>0,a,c$ and $e$ are known constants and $\Phi$ is the CDF of the standard Normal distribution.
5
votes
6answers
351 views

How to integrate $\displaystyle 1-e^{-1/x^2}$?

How to integrate $\displaystyle 1-e^{-1/x^2}$ ? as hint is given: $\displaystyle\int_{\mathbb R}e^{-x^2/2}=\sqrt{2\pi}$ If i substitute $u=\dfrac{1}{x}$, it doesn't bring anything: ...
1
vote
0answers
26 views

Battery between liftimes

Suppose that the operating lifetime of a certain type of battery is an exponential random variable with $\theta$ $= 2$ (measured in years). Find the probability that a battery of this type will have ...
1
vote
0answers
31 views

Lifetime of pdf disk

The pdf for the lifetime X, in years, of a Superstuff disk drive is given as follows: $f(x) = \begin{cases} 2/x^2 & \text{for } x\geq2\text{ } \\ 0 & \text{elsewhere} \end{cases}$. ...
2
votes
1answer
144 views

Structure of the functional space $\int_ {- \infty} ^ \infty f (x) dx = 1 $

Please, help me with studying of useful practical features of the following functional space: $$\int_{-\infty}^\infty f(x) \, dx = 1$$ For example: 1) What basis types are most convenient for ...
2
votes
1answer
141 views

How to show that the difference of two Gumbel distributed random variables follows a Logistic distribution?

How can you show that when you have two random variables $X,Y\sim\text{Gumbel}[0,1]$ , then $X-Y\sim\text{Logistic}[0,1]$ . I tried to use the convolution formula ...
1
vote
2answers
124 views

Integrating a special skew normal — the CDF of a convolution of a normal with a truncated normal

I am having a little trouble trying to compute an integral. In short, I wish to solve the following: $$F(x) = \int_{-\infty}^x \phi(au-b)\,\Phi(au+b)\,du $$ My intuition is that this might be ...
1
vote
0answers
23 views

Anti-derivative of a function involving exponentially distributed variable

Suppose a random variable $x$ with p.d.f $f(x) = \lambda e^{-\lambda x}$ such that $\lambda$ is the parameter of $f$. Given a function $ g(x) = (a + bx )e^{- \frac{\lambda x}{a}} $ where $a,b \in ...
1
vote
1answer
193 views

Finding the unconditional distribution

I found two similar questions. One has a good answer What is the distribution of an unconditioned random variable knowing the conditional distribution? . I have a similar problem that I think should ...
0
votes
1answer
86 views

expectation of logarithm under generalised inverse gaussian

I want to follow the following integral: $$\frac{1}{C}\int_0^\infty \log(z)\,z^{p-1}\exp\left(-\frac{az+b/z}{2}\right)\,dz$$ where C is the normalising constant. The following might be useful ...
1
vote
1answer
154 views

Does the following Integral have a Closed Form Solution?

Is there a solution to the following Integral that can get rid of the integral, but can leave the answer in terms of $\Phi()$ (the standard normal CDF) and $\phi $ (the standard normal PDF).? $ ...
0
votes
1answer
62 views

what can we say about $G(.)$?

Given $c \in R$, a deterministic probability density $f(x)$ and its cumulative distribution $F(c)$, what can be said about $G(c)$ where: $G(c)=\int f(x)F\left( x+c\right) dx $ The question ...
2
votes
1answer
646 views

Iterated Integrals and Unbounded Regions

Context I am having difficulty finding the posterior distribution of a Bayesian model with two parameters, which involves evaluating a double integral over an unbounded region. I prefer not to post ...