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.
True
False
A and C are independent.
True
False
A and B are independent given D .
True
False
A and C are independent given D .
True
False
Suppose three random variables X , Y , Z have a joint distribution
PX,Y,Z(x,y,z)=PX(x)PZ∣X(z∣x)PY∣Z(y∣z).
Then, X and Y are independent given Z .
True
False
Suppose random variables X and Y are independent given Z , then the joint distribution must be of the form
PX,Y,Z(x,y,z)=h(x,z)g(y,z),
where h,g are some functions.
True
False