EE 325
Homework 1 , Due date: Wed, 24th Jan
Total Marks: 15
General rules: It is ok if you discuss the homework with other students, but it is mandatory
to write the answers on your own and mention at the start who all you have discussed the
homework with. Failure to do so will lead to a straight rejection of the homework submission
and might incur further penalty.
Submission rule: Make a single PDF file with all your solutions and upload it on the moodle
submission portal before the deadline to avoid any penalty. Please try to maintain the order of
the questions (for example, solution to Q1 before Q2) while making the PDF.
1. Assuming that event X is independent of event Y and X is conditionally independent of
event Z given Y , show that X is independent of event Y ∩ Z. [2 marks]
2. Show that if two events A and B are independent, then Ac and B c are also independent.
[2 marks]
3. A coin is tossed independently n times. The probability of heads at each toss is p. At
each time k (k = 2, 3, ..., n), we get a reward if k th toss was a head and the previous toss
was a tail. Let Ak be the event that a reward is obtained at time k.
(a) Are events Ak and Ak+1 independent? [1 mark]
(b) Are events Ak and Ak+2 independent? [1 mark]
(c) Are events Ak and Ak+2 conditionally independent given Ak+1 ? [1 mark]
Justify all answers.
4. There are three types of coins which have different probabilities of landing heads when
tossed.
• Type A coins are fair, with probability 0.5 of heads
• Type B coins are biased and have probability 0.7 of heads
• Type C coins are biased and have probability 0.9 of heads
Suppose I have a drawer containing 5 coins: 2 of type A, 2 of type B, and 1 of type C. I
reach into the drawer and pick a coin at random. Without showing you the coin I flip it
once and get heads. What is the probability it is Type A? Type B? Type C? [3 marks]
5. Prove that
(a) P(A ∩ B) ≥ P(A) + P(B) − 1 [2 Marks]
(b) P(A1 ∩ A2 ∩ · · · ∩ An ) ≥ P(A1 ) + P(A2 ) + · · · + P(An ) − (n − 1) [ 3 Marks]
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