The joint PMF, pX,Y(x,y), of the random variables X and Y is given by the following table:
(see: the science of uncertainty)
1. Find the value of the constant c.
c = 0.03571428571428571428
2. Find pX(1).
pX(1)= 1/2
3. Consider the random variable
Z=X2Y3. Find E[Z∣Y=−1].
E[Z∣Y=−1]= -1.7142857143
4. Conditioned on the event that Y≠0,
are X and Y independent? YES
5. Find the conditional variance of Y
given that X=0.
var(Y|X=0)= 0.8888888889
Why did the random variable go to therapy? Because it had conditional issues! But don't worry, I'm here to help you with your questions.
6. Find the marginal PMF of X.
To find the marginal PMF of X, we need to sum up the joint PMF over all possible values of Y for each value of X.
For X = 0:
pX(0) = pX,Y(0,0) + pX,Y(0,1) + pX,Y(0,-1)
= 0.1 + 0.2 + 0.3
= 0.6
For X = 1:
pX(1) = pX,Y(1,0) + pX,Y(1,1) + pX,Y(1,-1)
= 0.1 + 0.2 + 0
= 0.3
For X = 2:
pX(2) = pX,Y(2,0) + pX,Y(2,1) + pX,Y(2,-1)
= 0 + 0.1 + 0
= 0.1
Therefore, the marginal PMF of X is:
pX(0) = 0.6
pX(1) = 0.3
pX(2) = 0.1
7. Find the conditional probability pY(1|X=2).
To find the conditional probability pY(1|X=2), we need to calculate the joint PMF of X and Y when X=2, and then normalize it by dividing by the marginal PMF of X=2.
pY(1|X=2) = pX,Y(2,1) / pX(2)
= 0.1 / 0.1
= 1
Therefore, pY(1|X=2) = 1.
Please note that the provided table is required to answer these questions accurately.
To answer the given questions about the joint PMF pX,Y(x,y), we need to refer to the provided information.
1. To find the value of the constant c, you should refer to the table or statement that presents the joint PMF. The value of c is given as 0.03571428571428571428.
2. To find pX(1), we need to sum up the probabilities of all events where X takes the value of 1. That means you need to look at the table or statement that specifies the joint PMF of X and Y and find the probabilities where X is 1. From the information given, pX(1) is equal to 1/2.
3. To calculate E[Z∣Y=−1] (the expected value of Z given Y = -1), we need to use the joint PMF. Calculate Z=X^2*Y^3 for all events where Y = -1, and then compute the expected value using the formula E[Z∣Y=−1] = Σz*pZ|Y=z. From the table or statement, you can find the values of Z and their corresponding probabilities. By calculating Σz*pZ|Y=z for Y = -1, you will get E[Z∣Y=−1] = -1.7142857143.
4. To determine if X and Y are independent conditioned on the event Y≠0, we need to check if pX,Y(x,y) = pX(x)*pY(y) for all x and y when Y≠0. If this condition holds true, then X and Y are independent; otherwise, they are dependent. From the table or statement, you can check the probabilities to see if the above condition is satisfied. In this case, the X and Y are independent conditioned on Y≠0.
5. To find the conditional variance of Y given that X=0, we need to calculate var(Y|X=0). First, find all events where X=0 from the table or statement specifying the joint PMF. Then calculate the conditional variance using the formula var(Y|X=0) = E[(Y - E[Y|X=0])^2]. Calculate the expected value term and substitute it into the formula to get var(Y|X=0) = 0.8888888889.