# 10.3 Problems on functions of random variables

Suppose X is a nonnegative, absolutely continuous random variable. Let $Z=g\left(X\right)=C{e}^{-aX}$ , where $a>0,\phantom{\rule{0.277778em}{0ex}}C>0$ . Then $0 . Use properties of the exponential and natural log function to show that

${F}_{Z}\left(v\right)=1-{F}_{X}\left(-,\frac{ln\left(v/C\right)}{a}\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}0

$Z=C{e}^{-aX}\le v$ iff ${e}^{-aX}\le v/C$ iff $-aX\le ln\left(v/C\right)$ iff $X\ge -ln\left(v/C\right)/a$ , so that

${F}_{Z}\left(v\right)=P\left(Z\le v\right)=P\left(X\ge -ln\left(v/C\right)/a\right)=1-{F}_{X}\left(-,\frac{ln\left(v/C\right)}{a}\right)$

Use the result of [link] to show that if $X\sim$ exponential $\left(\lambda \right)$ , then

${F}_{Z}\left(v\right)={\left(\frac{v}{C}\right)}^{\lambda /a}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}0
${F}_{Z}\left(v\right)=1-\left[1,-,e,x,p,\left(-,\frac{\lambda }{a},·,ln,\left(v/C\right)\right)\right]={\left(\frac{v}{C}\right)}^{\lambda /a}$

Present value of future costs. Suppose money may be invested at an annual rate a , compounded continually. Then one dollar in hand now, has a value ${e}^{ax}$ at the end of x years. Hence, one dollar spent x years in the future has a present value ${e}^{-ax}$ . Suppose a device put into operation has time to failure (in years) $X\sim$ exponential $\left(\lambda \right)$ . If the cost of replacement at failure is C dollars, then the present value of the replacement is $Z=C{e}^{-aX}$ . Suppose $\lambda =1/10$ , $a=0.07$ , and $C=1000$ .

1. Use the result of [link] to determine the probability $Z\le 700,\phantom{\rule{0.277778em}{0ex}}500,\phantom{\rule{0.277778em}{0ex}}200$ .
2. Use a discrete approximation for the exponential density to approximate the probabilities in part (a). Truncate X at 1000 and use 10,000 approximation points.
$P\left(Z\le v\right)={\left(\frac{v}{1000}\right)}^{10/7}$
v = [700 500 200];P = (v/1000).^(10/7) P = 0.6008 0.3715 0.1003tappr Enter matrix [a b]of x-range endpoints [0 1000] Enter number of x approximation points 10000Enter density as a function of t 0.1*exp(-t/10) Use row matrices X and PX as in the simple caseG = 1000*exp(-0.07*t); PM1 = (G<=700)*PX' PM1 = 0.6005PM2 = (G<=500)*PX' PM2 = 0.3716PM3 = (G<=200)*PX' PM3 = 0.1003

Optimal stocking of merchandise. A merchant is planning for the Christmas season. He intends to stock m units of a certain item at a cost of c per unit. Experience indicates demand can be represented by a random variable $D\sim$ Poisson $\left(\mu \right)$ . If units remain in stock at the end of the season, they may be returned with recovery of r per unit. If demand exceeds the number originally ordered, extra units may be ordered at a cost of s each. Units are sold at a price p per unit. If $Z=g\left(D\right)$ is the gain from the sales, then

• For $t\le m,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}g\left(t\right)=\left(p-c\right)t-\left(c-r\right)\left(m-t\right)=\left(p-r\right)t+\left(r-c\right)m$
• For $t>m,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}g\left(t\right)=\left(p-c\right)m+\left(t-m\right)\left(p-s\right)=\left(p-s\right)t+\left(s-c\right)m$

Let $M=\left(-\infty ,m\right]$ . Then

$g\left(t\right)={I}_{M}\left(t\right)\left[\left(p-r\right)t+\left(r-c\right)m\right]+{I}_{M}\left(t\right)\left[\left(p-s\right)t+\left(s-c\right)m\right]$
$=\left(p-s\right)t+\left(s-c\right)m+{I}_{M}\left(t\right)\left(s-r\right)\left(t-m\right)$

Suppose $\mu =50\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}m=50\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}c=30\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}p=50\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}r=20\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}s=40.$ .
Approximate the Poisson random variable D by truncating at 100. Determine $P\left(500\le Z\le 1100\right)$ .

mu = 50; D = 0:100;c = 30; p = 50;r = 20; s = 40;m = 50; PD = ipoisson(mu,D);G = (p - s)*D + (s - c)*m +(s - r)*(D - m).*(D<= m); M = (500<=G)&(G<=1100); PM = M*PD'PM = 0.9209[Z,PZ] = csort(G,PD); % Alternate: use dbn for Zm = (500<=Z)&(Z<=1100); pm = m*PZ'pm = 0.9209

(See Example 2 from "Functions of a Random Variable") The cultural committee of a student organization has arranged a special deal for tickets to a concert. The agreement is that the organization will purchase tentickets at $20 each (regardless of the number of individual buyers). Additional tickets are available according to the following schedule: • 11-20,$18 each
• 21-30, $16 each • 31-50,$15 each
• 51-100, \$13 each

If the number of purchasers is a random variable X , the total cost (in dollars) is a random quantity $Z=g\left(X\right)$ described by

$g\left(X\right)=200+18{I}_{M1}\left(X\right)\left(X-10\right)+\left(16-18\right){I}_{M2}\left(X\right)\left(X-20\right)+\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}$
$\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left(15-16\right){I}_{M3}\left(X\right)\left(X-30\right)+\left(13-15\right){I}_{M4}\left(X\right)\left(X-50\right)$
$\text{where}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}M1=\left[10,\phantom{\rule{0.166667em}{0ex}}\infty \right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}M2=\left[20,\phantom{\rule{0.166667em}{0ex}}\infty \right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}M3=\left[30,\phantom{\rule{0.166667em}{0ex}}\infty \right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}M4=\left[50,\phantom{\rule{0.166667em}{0ex}}\infty \right)$

Suppose $X\sim$ Poisson (75). Approximate the Poisson distribution by truncating at 150. Determine $P\left(Z\ge 1000\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}P\left(Z\ge 1300\right)$ , and $P\left(900\le Z\le 1400\right)$ .

X = 0:150; PX = ipoisson(75,X);G = 200 + 18*(X - 10).*(X>=10) + (16 - 18)*(X - 20).*(X>=20) + ... (15 - 16)*(X- 30).*(X>=30) + (13 - 15)*(X - 50).*(X>=50); P1 = (G>=1000)*PX' P1 = 0.9288P2 = (G>=1300)*PX' P2 = 0.1142P3 = ((900<=G)&(G<=1400))*PX' P3 = 0.9742[Z,PZ] = csort(G,PX); % Alternate: use dbn for Zp1 = (Z>=1000)*PZ' p1 = 0.9288

Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive