1. Copying 1 billion (109) bits from a USB drive to your hard disk takes about
0.2 seconds under ideal conditions. However, every bit has a probability of roughly
p = 10 8 to be copied incorrectly. Using error correcting codes, your PC is able to
recognize and correct awed bits during the copying process. Assume that it takes
0:02 seconds to correct a awed bit. Use the Poisson approximation to calculate
the probability that copying a movie of 2.5 gigabytes takes less than 9 seconds.
2. Let X be a Poisson random variable with parameter
a) Which n = n( ) 0 is the most likely value of X, i.e. maximizes P(X = n)?
b) Suppose the experiment described by X has returned the value n 0. Which
parameter = (n) maximizes P(X = n)?
3. Suppose that the continuous RV X has c.d.f. given by
F(x) =8>:
0 if x 0
0 x 0
(a) Find the c.d.f. of X.
(b) What must be the value of c?
(c) Find EX.
(d) Compute E 1p1+X2 .
6. Let c> 0 and X Unif[0;c]. Show that the RV Y = c X has the same c.d.f.
and therefore also the same p.d.f. as X.
7. (a) Suppose that the duration T (in hours) of your morning routine (breakfast,
shower, etc.) is modeled by an exponential RV with parameter . You set your
alarm 1 hour before your bus leaves for UBC. For which value of do you have a
50% chance of catching the bus?
(b)* Calculate the nth moment of T