Find the probability mass function of X, the length of the longest run of consecutive heads in four flips of a fair coin.
Find the probability mass function of X, the length of the longest run of consecutive heads...
Question 2 In a series of 100 fair coin flips, on average, what is the longest consecutive run of either heads or tails? What about for 1000 fair coin flips? Perform a Monte Carlo simulation to answer this question. Hint: look at function rle(). For example, suppose in 10 coin flips we observe {H,H,H,T,T,H,T,T,T,T},{H,H,H,T,T,H,T,T,T,T}, then the longest run is four. use rstudio do this.
A fair coin is tossed until heads appears four times. a) Find the probability that it took exactly 10 flips. b) Find the probability that it took at least10 flips. c) Let Y be the number of tails that occur. Find the pmf of Y.
Let X equal to the number of heads after 4 flips of a fair coin? Derive the probability mass function for X, and plot it. Also, compute the E[X] of X.
A coin is tossed, what is the probability that two consecutive heads or two consecutive tails occur in at most four tosses? Draw a tree diagram first.
3. A coin is tossed, what is the probability that two consecutive heads or two consecutive tails occur in at most four tosses? Draw a tree diagram first. (6 points)
Problem 2. Consider n flips of a coin. A run is a sequence of consecutive tosses with the same result. For k<n, let Ek be the event that a run is completed at time k; this means that the results of the kth and (k1)st flips are different. For example, if 10 and the outcomes of the first 10 flips are HHHTTHHTTH then runs are completed at times 3,5,7,9 (a) Show that if the coin is fair, then the events...
Problem 4. Five coins are flipped. The first four coins will land on heads with probability 1/4. The fifth coin is a fair coin. Assume that the results of the flips are independent. Let X be the total number of heads that result Hint: Condition on the last flip. (a) Find P(X2) (b) Determine E[X] S.20
Problem 2. Consider n flips of a coin. A run is a sequence of consecutive tosses with the same result. For k 〈 n, let Ek be the event that a run is completed at time k; this means that the results of the kth and k1)st flips are different. For example, if n 10 and the outcomes of the first 10 flips are HHHTTHHTTH then runs are completed at times 3, 5,7,9 (a) Show that if the coin is...
Problem 3. 3. For a nonnegative integer-valued random variable X show that i-0 4. A coin comes up heads with probability p. It is flipped until two consecutive heads or two consecutive tails occur. Find the expected number of flips 5. Suppose that PX a)p, P[Xb-p, a b. Show that (X-b)/(a-b) is a Bernoulli variable, and find its variance 3. For a nonnegative integer-valued random variable X show that i-0 4. A coin comes up heads with probability p. It...
1. A fair coin is flipped four times. Find the probability that exactly two of the flips will turn up as heads. 2. A fair coin is flipped four times. Find the probability that at least two of the flips will turn up as heads. 3. A six-sided dice is rolled twice. Find the probability that the larger of the two rolls was equal to 3. 4. A six-sided dice is rolled twice. Find the probability that the larger of...