Download Problem Set #14 for ECE 313 at University of Illinois, Fall 1997 - Prof. Dilip Sarwate and more Assignments Statistics in PDF only on Docsity!
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1. The random variables X and Y have joint pdf given by
f X,Y (u,v) = {
2 exp –(u + v) 0 < u < v < ∞ 0 elsewhere. (a) Are the random variables X and Y independent? (b) Find P{ Y > 3 X }.
(c) For α > 0, find P{ X + Y ≤ α}. (d) Use the result in (c) to determine the pdf of the random variable Z = X + Y. (e) An alternative method for computing f Z (w) is to use the magical mystical integral formula
f Z (w) = (^) ∫
f (^) X,Y (u, w–u)du = (^) ∫
f (^) X,Y (w–v, v)dv. Use either integral to compute f Z (w) and
compare with the result obtained in part (d). (f) Find the pdf of max( X , Y ). Think before you write (which is a suggestion that you should consider taking to heart in general as well as in this particular case!).
2. Ross, #8, p.293 (p. 297 in the 4th edition)
3. Let ( X , Y ) have joint pdf f X , Y (u,v) =
c(1 – u (^2) +v (^2) ), 0 < u 2 +v (^2) < 1 , 0, otherwise. (a) What is the value of the constant c?
(b) Let α denote some number between 0 and 1. What are the values of P{ X^2 + Y^2 < α} and
P{ X^2 + Y^2 < α}?
(c) Find the pdf of the random variable R = X^2 + Y^2 and the pdf of the random variable Z =
R^2 = X^2 + Y^2. Be sure to specify the pdf for all values of the argument from –∞ to +∞. (d) Prove that your answers of part (c) are indeed valid pdfs.
4. Let ( X , Y ) be uniformly distributed on the interior of the square with vertices at (1,0), (0,1), (–1,0), and (0,–1). (a) Find the joint pdf of X + Y and X – Y. (b) Determine whether X and Y are uncorrelated or independent.
5. Let ( X , Y ) be uniformly distributed on the unit disc (radius = 1) centered at the origin of a two dimensional plane. Find the expected value of the distance from the origin to the point ( X , Y ).
6. X and Y are independent random variables uniformly distributed on [0,1]. (a) Find the pdf of Z = X + Y. (b) Let A = min( X , Y ) and B = max( X , Y ). Use the results on p. 146 of the Lecture Notes to
write down the joint pdf f A , B (α,β). You should get a pdf that we have studied in class except, of course, that α and β are being used in place of u and v. (c) In class, we also found the pdf of A + B when the joint pdf is as in part (b). Why is the pdf of A + B so remarkably similar to the pdf of Z?
7. In the Lecture Notes, it is alleged on page 146 that if X and Y are independent exponential
random variables with parameter 1, then W = X + Y and Z = X^2 + Y^2 have joint pdf
f W , Z (α,β) =
exp(–α)
2 β – α^2
, 0 < β < α < 2 β
0, elsewhere.
(a) Draw a sketch of the plane with axes α and β, and indicate the region over which the joint pdf is nonzero.
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(b) Explain why the joint pdf is zero for α < β.
(c) Find the marginal pdf of W by integrating f W , Z (α,β) with respect to β. (d) Use Proposition 3.1 (p. 266, or p. 271 in the 4th edition) of Ross to deduce what the pdf of W = X + Y ought to be, and compare the result to your answer of part (c). Are they the same or different? Explain.
8. Except for the trivial case when all the probability mass is at μ = E[ X ], there is probability
mass both to the left and right of μ; in particular, there is an ω ∈ Ω such that X (ω) < μ. Is it also true that if (E[ X ], E[ Y ]) = (μ 1, μ 2), then there is an ω ∈ Ω such that X (ω) < μ 1 and Y (ω) < μ 2? If you believe the result is true, prove it. Otherwise, give a counterexample to show that it is false.
9. Let the random variables X and Y be independent and uniformly distributed on (0,1). Find
E(| X – Y |) and Var( X – Y ).
1 0. Let E[ X ] = 1, E[ Y ] = 4, var( X ) = 4, var( Y ) = 9, and ρ X , Y = 0.
(a) If Z = 2( X + Y )( X – Y ), what is E[ Z ]? (b) If T = 2 X + Y and U = 2 X – Y , what is cov( T , U )? (c) If W = 3 X + Y + 2, find E[ W ] and var( W ). (d) If X and Y are jointly Gaussian random variables, and W is as defined in (c), what is P{ W > 0}?
11.(a) If var( X + Y ) = 36 and var( X – Y ) = 64, what is cov( X , Y )? If you are also told that
var( X ) = 3•var( Y ), what is ρ X , Y? (b) If instead of having values 36 and 64, var( X ) = var( Y ), are X and Y uncorrelated?
1 2. Two random break–points X (^) 1 and X (^) 2 are chosen on a stick of unit length, thereby
breaking the stick into three pieces. Thus, if X (^) 2 > X (^) 1, the three pieces have lengths X (^) 1, X (^) 2 – X (^) 1, and 1 – X (^) 2. (If X (^) 1 > X (^) 2, interchange X (^) 1 and X (^) 2 in the above. Luckily, P{ X (^) 1 = X (^) 2} = 0).
(a) Sketch the u–v plane and indicate on it the region(s) such that if the random point ( X (^) 1, X (^) 2)
lies in the region(s), then the pieces can form a triangle. (b) Assume that the random variables X (^) 1 and X (^) 2 are independent and uniformly distributed on
(0,1). Show that the desired probability is 1/4 in this case by integrating the joint pdf over the region(s) found in part (a). (c) Suppose that X (^) 1 is uniformly distributed on (0,1) and that the conditional density of X (^) 2
given X (^) 1 = u is uniform on (u,1), that is, we break the stick at X (^) 1 (choosing the break- point with uniform density), and then break the right-hand piece at X (^) 2 (choosing the break- point with uniform density on the right-hand piece). Let T denote the event that the pieces can form a triangle. Find the conditional probability P(T| X | = u), for 0 < u < 1. (Hint: you will find that different expressions apply depending on the value of u). Now find the unconditional probability using the result that
P(T) =
P(T| X (^) 1 = u)f X 1
(u)du
where the integral is over the range (–∞, ∞) in general (but only over (0,1) in this case). (d) With the same assumptions as in (c), now find the joint pdf of X (^) 1 and X (^) 2. Show that this
joint pdf is nonzero only over the triangular region 0 < u < v < 1. Now, use the method
