Intereting Posts

Multivariable Calculus books similar to “Advanced Calculus of Several Variables” by C.H. Edwards
Proofs for complete + totally bounded $\implies$ compact.
Sums of prime powers
How can I find the value of $\ln( |x|)$ without using the calculator?
Set of zeroes of the derivative of a pathological function
Finding the limit of $\left(\frac{n}{n+1}\right)^n$
Prove that the sequence converges
Different ways to come up with $1+2+3+\cdots +n=\frac{n(n+1)}{2}$
Show that $\lim_{n\rightarrow \infty} \sqrt{c_1^n+c_2^n+\ldots+c_m^n} = \max\{c_1,c_2,\ldots,c_m\}$
How can I prove $dz=dx+idy$?
First Pontryagin class on real Grassmannian manifold?
Product of all monic irreducibles with degree dividing $n$ in $\mathbb{F}_{q^n}$?
What's an Isomorphism?
Obtaining binomial coefficients without “counting subsets” argument
Proving that $\sigma_7(n) = \sigma_3(n) + 120 \sum_{m=1}^{n-1} \sigma_3(m)\sigma_3(n-m)$ without using modular forms?

This question asks about a variant of an alternating renewal process.

I am sitting at a cafe watching men and women walk by. The interarrival

time $X$ between successive men is iid with distribution $F$, while the interarrival time $Y$

between successive women is iid with distribution $G$. Both $F$ and $G$ are

nonlattice. Unlike the usual alternate renewal process, here we have two

independent renewal processes, one for each sex.

Is there a nice way to compute the asymptotic probability that the most recent person seen

is a man? (The usual theorem doesn’t quite apply, since the interarrival times

are within-gender only, not from person to person independent of gender.)

- First exit time for Brownian motion without drift
- Markov process vs. markov chain vs. random process vs. stochastic process vs. collection of random variables
- Is the distribution of an Ito diffusion at time t absolutely continuous wrt Lebesgue measure?
- What is more elementary than: Introduction to Stochastic Processes by Lawler
- What is the Fourier Transform for $\mathscr{F} \left\{\frac{\partial^2 (x^2p(x,t))}{\partial x^2} \right\}$ w.r.t $x$?
- Distribution of compound Poisson process

I’m interested both in general, and for the specific case that F and G are

gamma distributions…

- Recast the scalar SPDE $du_t(Φ_t(x))=f_t(Φ_t(x))dt+∇ u_t(Φ_t(x))⋅ξ_t(Φ_t(x))dW_t$ into a SDE in an infinite dimensional function space.
- Filtration of stopping time equal to the natural filtration of the stopped process
- Brownian motion introduction
- Solution of SDE $dX_t = \mu(t)X_tdt + \sigma X_t dW_t$
- stopped filtration = filtration generated by stopped process?
- Resource for Stochastic Calculus and Ito processes
- If $A$ is the generator of $(P_t)$, then $A+f$ is the generator of $(P_t^f)$
- Laplace transform of integrated geometric Brownian motion
- Independence of increments of some processes
- Hermite Polynomials and Brownian motion

First consider only one renewal process, describing i.i.d. interarrival times with integrable distribution $F$. Asymptotically, the distribution of the largest interval around the present time without any arrival is the size-biased transform of $F$ (more on this later) and the present time is uniformly distributed in this interval.

In other words, assume that $X$ is a random variable with distribution $F$. Then the size-biased transform of $F$ is the distribution of any random variable $\hat X$ such that

$$

E(h(\hat X))=\frac{E(Xh(X))}{E(X)},

$$

for every bounded measurable function $h$. And the time elapsed since the last arrival is asymptotically distributed like $U\hat X$, where $U$ is uniform on $(0,1)$ and independent on $\hat X$.

Now, consider two independent renewal processes, with respective integrable distributions $F$ and $G$. The asymptotic probability that the last event corresponds to the $F$ renewal process is

$$

p_F=P(U\hat X\le V\hat Y),

$$

where $U$, $\hat X$, $V$ and $\hat Y$ are independent, $U$ and $V$ are uniform on $(0,1)$, the distribution of $\hat X$ is the size-biased transform of $F$ and the distribution of $\hat Y$ is the size-biased transform of $G$.

Thus,

$$

p_F=\frac{E(XY;UX<VY)}{E(X)E(Y)},

$$

where $X$, $Y$, $U$ and $V$ are independent, $U$ and $V$ are uniform on $(0,1)$, the distribution of $X$ is $F$ and the distribution of $Y$ is $G$.

Our next task is to get rid of $U$ and $V$. To condition on $(X,Y)$, one needs to compute

$$

P(Ux<Vy)=[y<x]y/(2x)+[x<y](1-x/(2y)).

$$

Unless I made a mistake, this yields

$$p_F=\frac{E(Y^2;Y<X)+E(2XY-X^2;X<Y)}{2E(X)E(Y)}.$$

Note that this is only one among several algebraically equivalent formulas for $p_F$.

All the expectations in the formula for $p_F$ are integrals involving $F$, $G$ and the respective densities $f$ and $g$. For example,

$$

E(Y^2;Y<X)=E(Y^2(1-F(Y))=\int_0^{+\infty}y^2g(y)(1-F(y))\mathrm{d}y.

$$

**Post hoc checks:** Here are some properties of the result above, which hold and ought to, for the formula to make sense.

(1) One has $p_F+p_G=1$ (where $p_G$ is the result one gets interchanging $X$ and $Y$) and $p_F$ is obviously positive, hence $p_G$ is positive as well. This proves that $p_F$ is in $(0,1)$.

(2) If $F=G$, everything cancels out in the numerator except the $E(2XY;X<Y)$ term, hence $p_F=\frac12$.

(3) If $X$ is exponential with parameter $a$ and $Y$ is exponential with parameter $b$, $p_F=a/(a+b)$.

**Edit** Regarding check (3) above, for exponential $a$ and $b$ distributions, one can compute everything in the formula giving $p_F$ as a function of $a$ and $b$, starting with $E(X)=1/a$, $E(Y)=1/b$, $f(x)=a\mathrm{e}^{-ax}$, $F(x)=1-\mathrm{e}^{-ax}$, $g(y)=b\mathrm{e}^{-by}$, $G(y)=1-\mathrm{e}^{-by}$.

Furthermore, $E(Y^2;Y<X)=E(Y^2(1-F(Y))$ hence

$$

E(Y^2;Y<X)=\int_0^{+\infty}y^2b\mathrm{e}^{-by}\mathrm{e}^{-ay}\mathrm{d}y=\frac{2b}{(a+b)^3},

$$

by symmetry, $E(X^2;X<Y)=2a/(a+b)^3$, and finally,

$$

E(XY;X<Y)=\int_0^{+\infty}ax\mathrm{e}^{-ax}\int_x^{+\infty}by\mathrm{e}^{-by}\mathrm{d}y\mathrm{d}x=\int_0^{+\infty}ax\mathrm{e}^{-ax}(x+1/b)\mathrm{e}^{-bx}\mathrm{d}x,

$$

hence

$E(XY;X<Y)=2a/(a+b)^3+a/(b(a+b)^2)$. Simplifying everything yields the value of $p_F=a/(a+b)$ given above.

Or, one can remember that for exponential interarrival times the arrival times form Poisson processes, that the superposition of two independent Poisson processes with intensities $a$ and $b$ is itself a Poisson process with intensity $a+b$, and finally that each point in the resulting point process is either an $a$-point or a $b$-point, independently of everything else and with respective probabilities $a/(a+b)$ and $b/(a+b)$. Putting all this together, one sees directly why the last event is an $a$-event with probability $p_F=a/(a+b)$.

If I understand right: arrivals of men are independent from arrivals of women. And each one is a renewal process, with distributions $f(t)$ and $g(t)$ (i.e. $f(t)$ is the probability density for the times between successive men arrivals $X$).

We take a random time $t$ (large, so that initial conditions do not matter and we can assume homogeneity) and call $A$ the “age” of the men arrival process, i.e., the time interval from the last men arrival to time $t$ ; analogously, we call $B$ the “age” of the women arrival. We want to compute the probability that the most recent is a man, that is

$P(A < B)$

For this, we must compute the joint probability density of $A,B$. The marginal (density of the “age”) is given by

$$p(A) = \frac{1}{E(X)} \int_A^\infty f(t) dt$$

(this can be easily found by a limit argument; check eg here, page 119).

And the same for $B$. And $A,B$ are indepent, hence the probality seeked is computed integrating in the region $A>B$

- Collection of numbers always in increasing or decreasing order
- Determine and classify all singular points
- Complex numbers equation problem
- How many solutions does $\cos(97x)=x$ have?
- Upper/lower bound on covariance two dependent random random variables.
- Conditional convergence of Riemann's $\zeta$'s series
- Prove that $f(x)=d(x,A)=\inf_{y\in A}d(x,y)$ is continuous on $M$
- eqiuvalent norms in $H_0^2$
- Prove $\gcd(a+b,a^2+b^2)$ is $1$ or $2$ if $\gcd(a,b) = 1$
- Does this pattern have anything to do with derivatives?
- Finding the remainder from equations.
- Dual to the dual norm is the original norm (?)
- show that $\int_{-\infty}^{+\infty} \frac{dx}{(x^2+1)^{n+1}}=\frac {(2n)!\pi}{2^{2n}(n!)^2}$
- integrate $\int_0^{2\pi} e^{\cos \theta} \cos( \sin \theta) d\theta$
- Integration double angle