Products
Rewards 
from HOLOOLY

We are determined to provide the latest solutions related to all subjects FREE of charge!

Please sign up to our reward program to support us in return and take advantage of the incredible listed offers.

Enjoy Limited offers, deals & Discounts by signing up to Holooly Rewards Program

HOLOOLY 
BUSINESS MANAGER

Advertise your business, and reach millions of students around the world.

HOLOOLY 
TABLES

All the data tables that you may search for.

HOLOOLY 
ARABIA

For Arabic Users, find a teacher/tutor in your City or country in the Middle East.

HOLOOLY 
TEXTBOOKS

Find the Source, Textbook, Solution Manual that you are looking for in 1 click.

HOLOOLY 
HELP DESK

Need Help? We got you covered.

Chapter 2

Q. 2.10

Find the probability density function for the mixed random variable Y of Examples 2.3 and 2.5, having

F_{Y} (u) = 0.05 (u + 4) U (u) + (0.8 – 0.05 u) U (u – 10)

 

Step-by-Step

Verified Solution

Differentiating this cumulative distribution function gives

p_{Y} (u) = 0.05U (u)+0.05 (u+4) \delta (u) – 0.05 U (u – 10)+(0.8-0.05u) \delta (u-10)

which can be rewritten as

p_{Y} (u) = 0.05[U (u)-U (u – 10)]+0.2 \delta  (u)+0.3 \delta (u-`0)

Or

p_{Y} (u) = 0.05U (u) U (10 – u)]+0.2 \delta  (u)+0.3 \delta (u-`0)

but these final forms require using some properties of the Dirac delta function. In particular, 0.05(u + 4) \delta (u) = 0.05 \delta (u), because the terms are equal at u = 0 and both are zero everywhere else. Similarly, (0.8 -0.05 u) \delta (u – 10) = 0.3 \delta (u – 10) , because they match at the sole nonzero point of u = 10.