# Probability in terms of CDF and PDF

suppose we have the situation like: $$\text{Pr}\biggl(XY \leq \frac{\gamma_t(\mu_3Z+\mu_4)}{\zeta-\frac{\gamma_t \mu_1}{Y}-\frac{\gamma_t \mu_2}{X}}\biggr)$$, where $$X,Y,Z$$ are independent random variables having CDF $$F_X(x), F_Y(y), F_Z(z)$$ and PDF $$f_X(x), f_Y(y), f_Z(z)$$ and all other things are constant, then can we represent it like this:

$$\int_0^{\infty} \int_0^{\infty} \int_0^{\infty} \text{Pr}\biggl(XY \leq \frac{\gamma_t(\mu_3Z+\mu_4)}{\zeta-\frac{\gamma_t \mu_1}{Y}-\frac{\gamma_t \mu_2}{X}}\biggr)f_X(x) f_Y(y) f_Z(z)\text{d}x \text{d}y \text{d}z ?$$

Any help in this regard will be highly appreciated.

• I guess you're assuming $X,Y,Z$ are non-negative. Commented Jun 26, 2022 at 6:50
• Yes sir. Thank for your answer. Commented Jun 26, 2022 at 7:33

Well, yes but that's quite trivial, since the first factor inside the integral is constant and what you get is just $$1$$ times it. What you could do, is put the indicator of the event that's inside the $$Pr()$$ in there:

$$\mathbb{P}\biggl(XY \leq \frac{\gamma_t(\mu_3Z+\mu_4)}{\zeta-\frac{\gamma_t \mu_1}{Y}-\frac{\gamma_t \mu_2}{X}}\biggr) = \int_0^{\infty} \int_0^{\infty} \int_0^{\infty} \boldsymbol{1}_{xy \leq \frac{\gamma_t(\mu_3z+\mu_4)}{\zeta-\frac{\gamma_t \mu_1}{y}-\frac{\gamma_t \mu_2}{x}}} f_X(x) f_Y(y) f_Z(z)\text{d}x \text{d}y \text{d}z.$$

To solve the integrating regions solve the inequality. To simplify the constants, denote

• $$a = \frac{\mu_1}{\mu_3}$$
• $$b = \frac{\zeta}{\gamma_t \mu_3}$$
• $$c_1 = \frac{\mu_1}{\mu_3}$$
• $$c_2 = \frac{\mu_2}{\mu_3}$$

so the inequality becomes equivalent with

$$xy \leq \frac{z + a}{b-c_1/y - c_2/x}$$

Divide both sides by $$xy$$ to get

$$1 \leq \frac{z + a}{bxy - c_1x - c_2y}$$

I will assume $$a\geq 0$$. (If this isn't the case, you have to do it in two parts.) Then because $$z+a\geq 0$$, we must have that $$z+a \geq bxy - c_1x - c_2y > 0$$. From $$bxy - c_1x - c_2y > 0$$ we also get $$(bx-c_2)y>c_1x$$. I will also assume $$c_1\geq 0$$ so we must have $$bx>c_2$$ and $$y>\frac{c_1x}{bx-c_2}$$. (If $$c_1<0$$, the inequalities flip.)

We're assuming $$X, Y, Z$$ are exponential. Let them have rates $$\lambda_X, \lambda_Y, \lambda_Z$$, respectively. So we can write the integral as

$$\lambda_X \lambda_Y \lambda_Z \int_0^\infty \int_0^\infty \int_0^\infty \boldsymbol{1}_{xy\leq \frac{z+a}{b-c_1x-c_2y}} e^{-\lambda_X x -\lambda_Y y -\lambda_Z z }dz dy dx \\ = \lambda_X \lambda_Y \lambda_Z \int_\frac{c_2}{b}^\infty \int_{\frac{c_1x}{bx-c_2}}^\infty \int_{\max(0, bxy-c_1x-c_2y-a)}^\infty e^{-\lambda_X x -\lambda_Y y -\lambda_Z z }dz dy dx \\ = \lambda_X \lambda_Y \int_\frac{c_2}{b}^\infty \int_{\frac{c_1x}{bx-c_2}}^\infty e^{-\lambda_X x -\lambda_Y y}e^{-\lambda_Z \max(0, bxy-c_1x-c_2y-a)} dydx \\ = \lambda_X \lambda_Y \left( \int_\frac{c_2}{b}^\infty \int_{\frac{c_1x}{bx-c_2}}^\frac{c_1x+a}{bx-c_2} e^{-\lambda_X x -\lambda_Y y} dydx + \int_\frac{c_2}{b}^\infty \int_{\frac{c_1x+a}{bx-c_2}}^\infty e^{-\lambda_X x -\lambda_Y y-\lambda_Z(bxy-c_1x-c_2y-a)} dydx \right)$$

But by simulation I get that this is wrong! The following SageMath simulation gives approximately $$0.35$$ (see values for the constants in the code)

def randExpo(l):
return -float(log(1-random()))/l

def simu(a,b,c1,c2, ls):
x,y,z = (randExpo(l) for l in ls)
return 1 if x*y <= (z+a)/(b*x*y-c1*x-c2*y) else 0

a = 1.2
b = 3.1
c1 = 0.7
c2 = 0.35
ls = (1.3, 0.77, 2.34) #the rates for X, Y, Z
simuN = 10000
print (float(sum(simu(a,b,c1,c2,ls) for _ in range(simuN))/simuN))


But calculating the intgral here in Desmos gives the value $$0.23$$. Where is my error?

• Thank you sir for your response. Say, $X,Y,Z$ are independent exponential random variables then how to proceed with the above integration? I am getting confused due to $\leq$ symbol. Commented Jun 26, 2022 at 7:48
• You're welcome. You would have to solve the inequality to see over which region(s) you have to integrate. I would first divide by $xy$ (you can forget the case $xy=0$, it has zero prob). Then split into two cases depending whether the denominator is positive/negative. Multiply the denominator to other side and you should get something nice in terms of $z$. Commented Jun 26, 2022 at 7:57
• Ok sir. Could please write mathematically. Commented Jun 26, 2022 at 8:01
• Thank you sir. But it has created more confusion now. Commented Jun 26, 2022 at 9:22
• @chaaru No problem. I've tried to make it more clear. I renamed the constants (I don't use $d$ anymore in order to not mix with the integration $dx, dy$). But I'm getting an error! The value I get isn't confirmed by simulation. Can someone spot where I made a mistake? Commented Jun 26, 2022 at 15:04