Problem 1
a) Independence in pairs does not imply the multiplication rule for events. Suppose we
toss a die twice and let A=”odd outcome of the first toss”, B=”odd outcome of the second
toss”, C=”sum is odd”. Show that there is a pairwise independence, i.e., A and B, B and C
as well as A and C are independent by showing
P(A, B) = P(A)P(B), P (B, C ) = P(B)P(C), P (A, C) = P(A)P(C).
Further, show that A,Band C(all three) are not independent by showing
P(A, B, C )6=P(A)P(B)P(C).
b) Multiplication rule does not imply independence in pairs
We will show that
P(A, B, C ) = P(A)P(B)P(C) (1)
does not imply that A, B and C are independent. Let the sample space be S={1,2,3,4,5}
and let A={1,4}, B={2,4}, C={3,4}and P r(1) = P r(2) = P r (3) = p−p3, P r(4) = p3,
where p= 0.1. Show that A,B and C satisfy (1) while P(A, B ) = p3, i.e, A and B are not
independent.
Problem 2.
Write a matlab code to simulate 1000 random variables with probability distribution
f(x) = 5x4
on the interval [0,1]. Calculate the numerical cumulative distribution of the simulated ran-
dom variables by counting how many of them are less than a given number. Plot the
numerical cumulative distribution against analytical one obtained from f(x).
Problem 3
Use rejection method to simulate 1000 random variable with probability distribution
f(x) = 1
2√πexp(−x2/2)
on the interval [0,∞] using g(x) = exp(−x). Calculate the numerical cumulative distribution
of the simulated random variables by counting how many of them are less than a given
number. Plot the numerical cumulative distribution against analytical one. Use erf function
for the analytical distribution.
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