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
Let's go step by step to determine whether the given statements are true or false.
1. A and B are independent.
To determine if A and B are independent, we need to compare the probabilities of their individual occurrences to the probability of both events occurring together. In this case, A represents the first coin toss being H, and B represents the second coin toss being H. Since the coin chosen is either fair or two-headed with equal probability, the probability of getting H in the first toss is 0.5, regardless of the chosen coin. Similarly, the probability of getting H in the second toss is also 0.5, regardless of the chosen coin. Therefore, A and B are independent.
Answer: True
2. A and C are independent.
Event C represents getting HH in two consecutive coin tosses. To determine if A and C are independent, we need to compare the probabilities of their individual occurrences to the probability of both events occurring together. In this case, A represents the first coin toss being H, and C represents getting HH in two consecutive coin tosses. However, the probability of getting HH depends on the chosen coin. If the two-headed coin is chosen, the probability of getting HH is 1. If the fair coin is chosen, the probability of getting HH is 0.5 * 0.5 = 0.25. Since the probability of event C changes depending on the chosen coin, A and C are not independent.
Answer: False
3. A and B are independent given D.
Event D represents the fair coin being chosen. To determine if A and B are independent given D, we need to compare the probabilities of their individual occurrences given the condition D to the probability of both events occurring together given D. In this case, both A and B are independent of the choice of coin given D because D represents the fair coin being chosen. Since the chosen coin is fair, the probability of getting H in the first toss and the probability of getting H in the second toss are both 0.5. Therefore, A and B are independent given D.
Answer: True
4. A and C are independent given D.
To determine if A and C are independent given D, we need to compare the probabilities of their individual occurrences given the condition D to the probability of both events occurring together given D. In this case, since D represents the fair coin being chosen, the events A and C are independent given D because the probability of event C (getting HH) does not depend on the outcome of event A (first coin toss being H) when the fair coin is chosen. Therefore, A and C are independent given D.
Answer: True
5. X and Y are independent given Z.
In general, if three random variables X, Y, and Z have a joint distribution described as PX,Y,Z(x, y, z) = PX(x)PZ|X(z|x)PY|Z(y|z), then X and Y are independent given Z. Therefore, X and Y are independent given Z.
Answer: True
6. X and Y being independent given Z implies a specific form of the joint distribution.
In general, if X and Y are independent given Z, then the joint distribution PX,Y,Z(x, y, z) can be factorized as PX,Y,Z(x, y, z) = h(x, z)g(y, z), where h and g are some functions. Therefore, X and Y being independent given Z implies a specific form of the joint distribution.
Answer: True