Let X, Y, Z, be independent discrete random variables.
Let A= X(Y+Z) and B= XY
With A, B, X, defined as before, determine wheter the folllowing statements are true or false.
1. A and B are independent
2. A and B are conditionally independent, given X = 0.
3. A and B are conditionally independent, given X = 1.
Hey P,
did you found the answers for this question?
Well, well, well. Let's take a look at these statements.
1. A and B are independent.
Hmm, the random variables A and B can only be independent if their joint distribution equals the product of their individual distributions. Since A = X(Y + Z) and B = XY, it's clear that they are not independent because A depends on both Y and Z, while B only depends on X and Y. So, statement 1 is FALSE.
2. A and B are conditionally independent, given X = 0.
To answer this, we need to consider the definition of conditional independence. If two random variables are conditionally independent given a certain condition, their joint distribution, given that condition, should be equal to the product of their individual conditional distributions. In this case, given X = 0, A = 0 and B = 0. Since both A and B are always equal to 0 when X = 0, their joint distribution is indeed equal to the product of their individual conditional distributions. Therefore, statement 2 is TRUE. Ah, conditional independence strikes again!
3. A and B are conditionally independent, given X = 1.
Now, for this statement to be true, the joint distribution of A and B given X = 1 should be equal to the product of their individual conditional distributions. However, given X = 1, A = (Y + Z) and B = Y. Since A and B clearly depend on both Y and Z, their joint distribution, given X = 1, cannot be written as the product of their individual conditional distributions. So, statement 3 is FALSE.
Hope that clears things up for you!
To determine whether the statements are true or false, we need to analyze the relationship between the random variables X, Y, Z, A, and B.
1. A and B are independent:
Two random variables are independent if and only if the joint probability distribution is the product of the marginal probability distributions. In this case, A = X(Y + Z) and B = XY.
To check independence, we need to compare the joint probability distribution P(A=a, B=b) with the product of the marginal probability distributions P(A=a) * P(B=b). Without any information about the probability distributions of X, Y, and Z, it is not possible to determine whether A and B are independent or not. Therefore, the answer to this statement is insufficient information.
2. A and B are conditionally independent, given X = 0:
For A and B to be conditionally independent given X = 0, the conditional probability distribution P(A=a | X=0, B=b) should be equal to P(A=a | X=0) for all possible values of a and b.
However, since A = X(Y + Z), if X = 0, then A = 0 regardless of the values of Y and Z. So, the conditional probability P(A=0 | X=0, B=b) would depend only on B, not on any information about Y and Z.
Therefore, the statement is false. A and B are not conditionally independent given X = 0.
3. A and B are conditionally independent, given X = 1:
Similar to the previous statement, for A and B to be conditionally independent given X = 1, the conditional probability distribution P(A=a | X=1, B=b) should be equal to P(A=a | X=1) for all possible values of a and b.
Since A = X(Y + Z), if X = 1, then A = Y + Z. In this case, the value of B = XY is determined by X. Therefore, knowing B provides no additional information about A when X = 1.
Thus, P(A=a | X=1, B=b) = P(a | X=1), and A and B are conditionally independent given X = 1.
So, the answer to this statement is true. A and B are conditionally independent given X = 1.
To determine whether the statements are true or false, we need to examine the definitions of independence and conditional independence for random variables.
1. A and B are independent:
Two random variables A and B are considered independent if and only if the joint probability distribution function can be factored into the product of their marginal probability distribution functions. In other words, the values of one variable do not provide any information about the values of the others.
To determine if A and B are independent, we need to compare their joint probability distribution, P(A, B), with the product of their marginal probability distributions P(A) * P(B). If they are equal, A and B are independent; otherwise, they are dependent.
2. A and B are conditionally independent, given X = 0:
Two random variables A and B are considered conditionally independent given the value of another random variable X = x if and only if the conditional probability distribution of A given X = x is equal to the marginal probability distribution of A and vice versa.
To determine if A and B are conditionally independent given X = 0, we need to compare the conditional probability distribution of A given X = 0, P(A | X = 0), with the marginal probability distribution of A, P(A). If they are equal, A and B are conditionally independent given X = 0; otherwise, they are dependent under this condition.
3. A and B are conditionally independent, given X = 1:
Similarly, to determine if A and B are conditionally independent given X = 1, we need to compare the conditional probability distribution of A given X = 1, P(A | X = 1), with the marginal probability distribution of A, P(A).
To give a final verdict on the truth or falsity of each statement, we need to evaluate these comparisons based on the specific details of the random variables X, Y, and Z and their probability distributions.