Suppose that we have a box that contains two coins:

A fair coin: P(H)=P(T)=0.5 .

A two-headed coin: P(H)=1 .

A coin is chosen at random from the box, i.e. either coin is chosen with probability 1/2 , and tossed twice. Conditioned on the identity of the coin, the two tosses are independent.

Define the following events:

Event A : first coin toss is H .

Event B : second coin toss is H .

Event C : two coin tosses result in HH .

Event D : the fair coin is chosen.

For the following statements, decide whether they are true or false.

A and B are independent.

A and C are independent.

A and B are independent given D .

A and C are independent given D .

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Conditional Independence 2
2 points possible (graded, results hidden)
Suppose three random variables X , Y , Z have a joint distribution

Then, X and Y are independent given Z .

Suppose random variables X and Y are independent given Z , then the joint distribution must be of the form

where h,g are some functions.


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