FinMath September Review 2013
Exercises for Greg Lawler’s third lecture
I will be posting solutions to all of my problems some time next week. Feel free to bug
me by e-mailing me (lawler@math.uchicago.edu) if I forget.
Also, you can download a copy of my stochastic calculus notes/book which I will be using
this year at
www.math.uchicago.edu/ lawler/finbook.pdf
This is not to be shared with others but you may use it. Sections 1.1 and 2.2 are relevant
for Wednesday’s lecture.
Exercise 1. Let X1 , X2 , . . . be independent standard normal distributions with mean µ and
variance σ 2 . Let
n
1X
X1 + · · · + Xn
2
, Sn =
[Xj − X̄n ]2 .
X̄n =
n
n j=1
Give the distributions of the following random variables (including the values of the parameters).
X̄n
Sn
σ2
n
1 X
(Xj − µ)2 ,
2
σ j=1
√
n X̄n
Sn
Exercise 2. Let X, Y denote the values of two indepedent die rolls (so each takes values
1, ..., 6), and let Z = X + Y and let W = 1 or 2 depending on whether Z is even or odd.
Give the following conditional expectations. In each case, be explicit about how many different
values it takes.
E[X | Y ]
E[X + Z | Y ]
E[Z | W ]
E[Y | W ]
E[Z | X + Y ]
E[W | Y ].
E[3X 2 + 2X | X]
Aree Y and Z independent random variables? Are Y and W independent random variables?
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Exercise 3. In the previous exercise find Cov(X, Z), Cov(X, Y ), Cov(Z, W ).
Exercise 4. Suppose Z1 , Z2 , Z3 are independent standard normals and
X = 2Z1 + 3Z2 + 4Z3 ,
Y = Z1 − Z3 ,
W = X − Y.
Why does (W, X, Y ) have a joint normal distribution? Find the covariance matrix. Are W, X
independent? Are W, Y independent?
Exercise 5. In the previous example, let R = Z1 − Z2 , Q = Z1 + Z2 . Are R and Q independent?
Exercise 6. Find two random variables X, Y that are not independent that satisfy E[XY ] =
E[X] E[Y ]. (They better not have a joint normal distribution!)
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