18.440 — Review Problems for Quiz 2
Sunday, April 15, 2005
1. (a) (5pts) State the DeMoivre-Laplace theorem (the central limit theorem for the binomial distributions).
(b) (15pts) Let X be the number of times that a fair coin, flipped 1600 times, lands
heads. Use the normal approximation to find P(760 ≤ X ≤ 840). (You do not have to
use the continuity correction.)
Table: Area Φ(x) under the standard normal curve to the left of x.
x
Φ(x)
0.5
.6915
1.0
.8413
1.5
.9332
2.0
.9772
2.5
.9938
3.0
.9987
2. Customers enter a bank at a Poisson rate of λ for every minute. Let X(t) be the number
of customers entering a bank for t minutes. Find with a brief explanation
(a) (10pts) P(X(t) ≤ 1);
(b) (10pts) E X(t)(X(t) − 1)(X(t) − 2) · · · (X(t) − 2004) .
3. Chapter 5 problems #10, #25, #37
4. Chapter 6 problems #9, #22 (add the following parts)
9(e) Find fY |X (y|x) and the conditional density of Y given x = 34 , in other words, find
fY |X (y|x) = 34 ).
9(f) Find the joint density of X, Y conditioned on the event A : 0 ≤ x ≤ 12 , 0 ≤ y ≤ 21 ,
i.e. find fX,Y |A (x, y|A).
22(d) Find fX|Y (x|y) and the conditional density of X given y = 13 , in other words, find
fX|Y (x|y) = 31 ).
22(e) Find the joint density of X, Y conditioned on the event A : y ≤
fX,Y |A (x, y|A).
1
1
2,
i.e. find