Additional Problem 2. Two fair dice are thrown. Let Xi and X2 denote the outcomes pint...
3. Two fair, four-sided dice are rolled. Let X1, X2 be the outcomes of the first and second die, respectively. (a) Find the conditional distribution of X2 given that Xi + X2 = 4. (b) Find the conditional distribution of X2 given that Xi + X2-5.
2. Let X1 and X2 be the numbers showing when two fair dice are thrown. Define new random variables X = Xi-X2 and Y = X1 + X2. Show that X and Y are uncorrelated but not independent. Hint: To show lack of independence, it is enough to show that PlX = j, Y = k]メPIX = j] . PY = k] for one pair (j, k); try the pair (0.2).]
2. Let X1 and X2 be the numbers showing when two fair dice are thrown. Define new random variables XX1 - X2 and Y -X1 + X2. Show that X and Y are uncorrelated but not independent. Hint: To show lack of independence, it is enough to show that PX = j, Y = k]メPX = j] . P[Y = 서 for one pair (j, k); try the pair (0.2).]
1. Toss two fair dice and let E1 denote the event that the sum
is 10 and E2 the event that the first is 3. Show that the two
events are not independent. Construct two independent events.
Construct 4 events that are not independent but such that they are
pairwise independent.
1. Toss two fair dice and let E1 denote the event that the sum is 10 and E2 the event that the first is 3. Show that the two...
A pair of fair dice is tossed. Let X denote the larger of the two numbers showing. Find the expected value of X.
3. Two fair dice are thrown. Let X be the smaller of the two numbers obtained (or the common value if the same number is obtained on botih dice). Find the probability mass function of X. Find P(X>3).
1. Two dice are thrown. Let E be the event that the sum of the dice is odd, let F be the event that at least one of the dice lands on 1, and let G be the event that the sum is 5 . List the outcomes ina. E ∩ Fb. E ∪ Fc. E ∩ F'd. E ∩ F ∩ G
Q1 (100) Suppose you roll two twenty-five-sided dice. Let X1, X2 the outcomes of the rolls of these two fair dice which can be viewed as a random sample of size 2 from a uniform distribution on integers. a) What is population from which these random samples are drawn? Find the mean (u) and variance of this population (62)? Show your calculations and results.
Two fair 6-sided dice are tossed. Let X denote the number appearing on the first die and let y denote the number appearing on the second die. Show that X, Y are independent by showing that P(X = x, Y = y) = P(X = x) x P(Y = y) for all (x,y) pairs.
Two fair dice are thrown. What is the probability of at least one odd number? What is the probability of this if four fair dice are thrown?