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University Problem Set #13 ECE 313
of Illinois Page 1 of 2 Fall 2001
Assigned: Wednesday, November 14, 2001 Due: Wednesday, November 28, 2001 Reminder: Happy Thanksgiving Reading: Ross, Chapter 6 and Chapter 7 Noncredit Exercises: Ross, Chapter 6: Problems 26, 28-30, 41-43, 51, 54; Theoretical Exercises: 8, 14, 22, 23, 33; Chapter 7: Problems 1, 16, 26, 29, 34, 36; Theoretical Exercises: 1, 2, 17, 22, 23, 40 Problems:
1. Let ( X , Y ) have joint pdf f X , Y (u, v) =
C 1–u (^2) –v (^2) , u (^2) +v (^2) < 1 , 0 , elsewhere. (a) What is the value of C?
(b) Find P{ X^2 + Y^2 < 0.25}.
2. The random point ( X , Y ) is uniformly distributed on the shaded region shown in the left- hand figure below. (a) Find the marginal pdf f X (u) of the random variable X. (b) Write down the marginal pdf f Y (v) of the random variable Y from your answer to part (b). (c) Find P{ X < Y < 2 X }.
(d) What is f X | Y (u|α), the conditional pdf of X given that Y = α, if α satisfies 0 < α < 1/2?
What is f X | Y (u|α), the conditional pdf of X given that Y = α, if α satisfies 1/2 < α < 1? Now, apply the theorem of total probability to compute the unconditional pdf of X from f X | Y (u|α). Do you get the same answer as in part (a)?
u
1
1
v
1v
I R
R
1
2
3. Two resistors are connected in series to a one-volt voltage source as shown in the right- hand diagram above. Suppose that the resistance values R 1 and R 2 (measured in ohms) are independent random variables, each uniformly distributed on the interval (0, 1). Find the pdf f I (a) of the current I (measured in amperes) in the circuit.
4. Let ( X , Y ) have joint pdf f X , Y (u, v) = {
2u, 0 < u < 1 , 0 < v < 1 , 0 , elsewhere. Find the pdf of Z = X^2 Y.
5. (Unbelievable but true: this problem is easier than it looks…).
(a) If X is N (0,σ^2 ), use the magic formula in Example 7b, Chapter 5.7 of Ross to show that
X^2 has gamma pdf with parameter (1/2,1/2σ^2 ).
(b) Now, suppose that X , Y , and Z are independent N (0,σ^2 ) random variables. Then X^2 ,
Y^2 , and Z^2 are independent gamma random variables with parameter (1/2,1/2σ^2 ). Use the
University Problem Set #13 ECE 313
of Illinois Page 2 of 2 Fall 2001
comment immediately following the proof of Proposition 3.1 (p. 267, 5th ed. or p. 262, 6th ed.) of Ross to state what the type of pdf of W = X^2 + Y^2 + Z^2 is, and write down explicitly the exact pdf. What is the numerical value of f W (5) if σ^2 = 4?
(c) Prove that E[ W ] = 3σ^2. If you actually evaluated an integral to get this answer instead of using LOTUS, shame on you! (d) In a physical application, X , Y , and Z represent the velocity (measured along three perpendicular axes) of a gas molecule of mass m. Thus, H = (1/2)m W is the kinetic energy of the particle, and an important axiom of statistical mechanics asserts that the average kinetic energy is E[ H ] = E[(1/2)m W ] = (1/2)mE[ W ] = (3/2)mσ^2 = (3/2)kT where k is Boltzmann’s constant and T is the absolute temperature of the gas in °K. (Note that the average energy is (1/2)kT per dimension.) Show that the kinetic energy H has the
Maxwell-Boltzmann pdf f H (β) = 2 π
(kT)–3/2^ β•exp(–β/kT), β > 0.
(e) V = W = X^2 + Y^2 + Z^2 is the “speed” of the molecule. Show that the pdf of V is
f V (γ) = 4 π
m 2kT
3/2γ2exp
2 2kT , γ > 0 cf. Theoretical Exercise 1 of Chapter 5.
(f) What is the average speed of the molecule?
6. The number of hours R that a student spends r eading about probability in preparation for the ECE 313 Final Examination and the number of hours S that the student spends s leeping can be modeled as random variables with joint probability density function
f R , S (x,y) =
K,^10 ≤^ x + y^ ≤^ 20, x^ ≥^ 0, y^ ≥^ 0,
0, otherwise. (a) What is the value of K? (b) What is the marginal pdf of R? (c) Unfortunately, the more the student tries to read about probability, the more confused the student gets. Also, the less the student sleeps, the more tired the student gets. As a result, the student’s percentage score T on the Final Exam is related to S and R via the equation T = 50 + 2.5( S – R ). Find the pdf of T. (d) Noncredit exercise: Should S have denoted s tudying and R denoted r esting instead?
