EEL 6545 – Random Processes –HW Topic #3 Instructor: Dr. Richard D. Gitlin, TA Mahmood Azhar (Fall 2014) Distinguished Professor of Electrical Engineering Phone: (813) 974-1321; Office: ENB 359 E-mail:
[email protected]
All assignments are due at the next class after the Topic is completed. Any exceptions need the prior approval of the instructor. PLEASE NOTE THAT ALL THE INTERMEDIATE STEPS NECESSARY TO SHOW THE COMPLETE SOLUTIONS ARE NOT GIVEN. STUDENTS ARE EXPECTED TO WORKOUT INTERMEDIATE PROBLEM DETAILS. For references to equations used, please see text book Papoulis: Chapter 4 ---problems 1. (Papoulis problem 4.6) We measure for resistance R of each resistor in a production line and we accept only the units the resistance of which is between 96 and 104 ohms. Find the percentage of the accepted units (a) if R is uniform between 95 and 105 ohms; (b) if R is normal with η=100 and σ=2 ohms.
2. (Papoulis problem 4.13) A fair coin is tossed three times and the random variable x equals the total number of heads. Find and sketch ( ) ( ).
3. (Papoulis problem 4.21) The probability of heads of a random coin is a random variable p uniform in the interval (0,1). (a) Find P {0.3≤p≤0.7}. (b) The coin is tossed 10 times and heads shows 6 times. Find the a posteriori probability that p is between 0.3 and 0.7.
4. (Papoulis problem 4.23) A fair coin is tossed 900 times. Find the probability that the number of heads is between 420 and 465.
5. (Papoulis problem 4.25)
If P(A)=0.6 and k is the number of successes of A in n trials (a) show that P {550 ≤ k ≤ 650}=0.999, for n=1000. (b) Find n such that P {0.59n ≤ k ≤ 0.61n}=0.95.
<- Table is 4-1 and n = 9126
Papoulis: Chapter 5 ---problems 6. (Papoulis problem 5.6) The random variable x is uniform in the interval (0,1). Find the density of the random variable y=-lnx.
7. (Papoulis problem 5.7) We place at random 200 points in the interval (0, 100). The distance from 0 to the first random point is a random variable z. Find ( ) (a) exactly and (b) using the Poisson approximation.
8. (Papoulis problem 5.12) The random variable x is uniform in the interval (-2π, 2 π). Find (c) y=2sin(3x+40o).
(y) if (a) y=x3 , (b) y=x4, and
9. (Papoulis problem 5.15) A fair coin is tossed 10 times and x equals the number of heads. (a) Find if y=(x-3)2.
( ). (b) Find
( )
10. (Papoulis problem 5.30) If x is uniform in the interval (10, 12) and y=x3 , (a) find (5-86).
(y); find E{y}: (i) exactly; (ii) using
Papoulis: Chapter 6 ---problems 11. (Papoulis problem 6.1) x and y are independent, identically distributed (i.i.d.) random variables with common p.d.f. ( )
( )
( )
( )
Find the p.d.f of the following random variables (a) x+y, (b)x-y, (c) xy, (d) x/y, (e) min(x,y), (f) max(x,y), (g) min (x,y)/max(x,y).
12. (Papoulis problem 6.4) The joint p.d.f of x and y is defined as: ( Define z=x-y. Find the p.d.f of z.
)
{
13. (Papoulis problem 6.12) x and y are independent uniformly distributed random variables on (0,1). Find the joint p.d.f. of (a) x+y and x-y.
14. (Papoulis problem 6.17) The random variables x and y are N(0,4) and independent. Find and (b) z=x/y.
( )
( ) if (a) z=2x+3y,
15. (Papoulis problem 6.58) The random variables x and y are jointly distributed over the region 0
)
{
For some k . Determine k. Find the variances of x and y. What is the covariance between x and y?
