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Problem 2 (5 points) Roll a fair die 5 times. Let X denote the number of...
Problem 2. (15 pts) A fair die is tossed 20 times in succession. Let Y be the total number of sixes that occur, and let X be the number of sixes occurring in the first 5 tosses. Determine the conditional probability mass function P(X r|Y ).
Problem 6. A fair die is rolled four times. (a) Let Y denote the number of distinct rolls. Find the probability mass function of Y. (b) Let Z denote the minimal result fo the 4 throws. Find the probability mass function of Z
Roll a fair die and denote the outcome by Y . Then flip Y many fair coins and let X denote the number of tails observed. Find the probability mass function and expectation of X.
problem 4 you repeatedly roll an ordinary eight-sided die five times. Let X equal the number of times you roll the die. Let Y equal the value of the first roll What is E[x] and E[Y]
Question : A fair die is tossed 20 times in succession. Let Y be the total number of sixes that occur, and let X be the number of sixes occurring in the first 5 tosses. Determine the conditional probability mass function P(X = x|Y = y).
In a dice game, you roll a fair die three times, independently. If you don’t roll any sixes, you lose 1 dollar. If you roll a six exactly once, you win one dollar. If you roll a six exactly twice, you win two dollars. If you roll a six all three times, you win k dollars. (A) Let k = 3. What is the expected value of the amount you would win by playing this game (rounded to the nearest...
2. Let X denote the outcome the outcome of a die roll. (i) Compute E[X] given that the roll is fair; (ii) construct a distribution function for X such that for all a (3.5,6)., E[x] a
Use a random number generator to simulate the roll of a fair die 100 times. Let the number face up on the die represent the variable X A. Build a relative frequency table of the outcomes of the variable X. X Freq Rel. Freq B. Use the relative frequency distribution from part c to estimate the probability of an even number face up, then find the actual probability using the probability distribution and comment on the difference in values.
4. Toss a fair coin 6 times and let X denote the number of heads that appear. Compute P(X ≤ 4). If the coin has probability p of landing heads, compute P(X ≤ 3) 4. Toss a fair coin 6 times and let X denote the number of heads that appear. Compute P(X 4). If the coin has probability p of landing heads, compute P(X < 3).
(3.) A fair six-sided die is rolled repeatedly. Let R denote the random variable representing the outcome of any particular roll. The following random variables are all discrete-time Markov chains. Specify the transition probabilities for each (as a check, make sure the row sums equal 1) (a) Xn, which represents the largest number obtained by the nth roll. (b) Yn, which represents the number of sixes obtained in n rolls.