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Problem 2. (6 pts) Independence and Conditional Probability (a) (2 pts) An urn contains 3 red...
(a) (2 pts) An urn contains 3 red and 5 green balls. At each step of this game, we pick one ball at random, note its color and return the ball to the urn together with anoter ball of the same color. Prove by induction that the probability that the ball we pick a red ball at the n-th step is 3/8. (b) (2pts) Consider any two random variables X, Y of any distirbution and not necesarily independent. Given that...
(a) (2 pts) An urn contains 3 red and 5 green balls. At each step of this game, we pick one ball at random, note its color and return the ball to the urn together with anoter ball of the same color. Prove by induction that the probability that the ball we pick a red ball at the n-th step is 3/8.
(a) (2 pts) An urn contains 3 red and 5 green balls. At each step of this game, we pick one ball at random, note its color and return the ball to the urn together with anoter ball of the same color. Prove by induction that the probability that the ball we pick a red ball at the n-th step is 3/8.
(a) (2 pts) An urn contains 3 red and 5 green balls. At each step of this game, we pick one ball at random, note its color and return the ball to the urn together with anoter ball of the same color. Prove by induction that the probability that the ball we pick a red ball at the n-th step is 3/8.
(e) (2 pts) Show that, if X and Y are two uncorrelated (ie. EXY = EXEY) Bernoulli (Indicator) random variables then they are independent.
EXEY) (c) (2 pts) Show that, if X and Y are two uncorrelated (i.e. EXY Bernoulli (Indicator) random variables then they are independent.
(b) (2pts) Consider any two random variables X, Y of any distirbution and not necesarily independent. Given that P(X = a) = P1, P(max(X,Y) = a) = P2, and P(min(X,Y) = a) = P3, find P(Y = a) in terms of P1, P2 and 23. Hint: Use the first Bayes theorem. Choose a suitable event-complement pair to partition the probability space by comparing X and Y.
Consider any two random variables X, Y of any distirbution and not necesarily independent. Given that P(X=a)=p1, P(max(X, Y) =a) =p2, and P(min(X, Y) =a) =p3. Find P(Y=a) in terms of p1, p2 and p3. Hint: Use the first Bayes theorem. Choose a suitable event-complement pair to partition the probability space by comparing X and Y .
Example. 2 urns. Red urn contains 3 red balls, 2 white balls. White urn contains 1 red ball, 4 white balls. Pick an urn randomly. Randomly select a ball from that urn. Without replacing the 1st ball, select a ball from the urn whose color matches the first ball. Q. Make a tree diagram, complete with probabilities describing this situation. Q. Find the probability that the first ball is white. Q. Find the probability that the second ball is white....
3. * One urn contains one black ball and one red ball. A second urn contains one white ball and one red ball. One ball is selected at random from each urn (a) Exhibit the sample space ? for this experiment. (b) Show the subset of ? that defines the event A that both balls will be of the same color. (c) Assume each elementary outcome in 2 is equally likely to occur. What is the probability that both balls...