![A fair coin is flipped independently until the first Heads is observed. Let the random variable K be the number of tosses until the first Heads is observed plus 1. For example, if we see TTTHTH, then K = 5. For k 1, 2, , K, let Xk be a continuous random variable that is uniform over the interval [0, 5]. The Xk are independent of one another and of the coin flips. LetX = Σ i Xo Find the mean and variance of X. You may use the fact that the mean and variance of a geometric random variable with parameter p are Ilp and (1-рмрг. respectively. Var(X)](http://img.homeworklib.com/questions/f6b4af20-55fd-11eb-a00a-d1c8641b3191.png?x-oss-process=image/resize,w_560)
Homework Answers
![Let the variable K be the number of tosses until the first Heads is observed, Since the coin is a fair, K follows a geometric distribution with parameter p- The PMF of K can be written as 2 The mean & variances of K are, P 1/2 2)12 Let Xbe a continuous random variable that is uniform over interval [0, 5] The mean & variances of Xt are, Elx, 5-0-5 (5-025 Var[X 12 12 Define a new random variable, x->x, に! Calculate the mean and variance of Y.](http://img.homeworklib.com/questions/f7ec41f0-55fd-11eb-bc9f-8fe216ddb655.png?x-oss-process=image/resize,w_560)
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A fair coin is flipped independently until the first Heads is observed. Let the random variable...
1. A fair coin is flipped until three heads are observed in a row. Let denote...
1. A fair coin is flipped until three heads are observed in a row. Let denote the number of trials in this experiment. [This is a simple model of some procedures in acceptance control]. b) Find p(x) for the first five values of X c) Make an estimate of EX. Hint: use geometric rv related to X.
6. A fair coin is flipped repeatedly until 50 heads are observed. What is the probability...
6. A fair coin is flipped repeatedly until 50 heads are observed. What is the probability that at least 80 flips are necessary? (You may calculate an approximate answer.)
Assume that a coin is flipped where the probability of coin lands "Heads" is 0.49. The...
Assume that a coin is flipped where the probability of coin lands "Heads" is 0.49. The coin is flipped once more. This time, the probability of obtaining the first flip's result is 0.38. The random variable X is defined as the total number of heads observed in two flips. On the other hand, the random variable Y is defined as the absolute difference between the total number of heads and the total number of tails observed in two flips. Calculate...
Exercise 8.52. A fair coin is flipped 30 times. LetX denote the number of heads among...
Exercise 8.52. A fair coin is flipped 30 times. LetX denote the number of heads among the first 20 coin flips and Y denote the number of heads among the last 20 coin flips. Compute the correlation coefficient of X and I.
Coin with random bias. Let P be a random variable distributed uniformly over [0, 1]. A...
Coin with random bias. Let P be a random variable distributed uniformly over [0, 1]. A coin with (random) bias P (i.e., Pr[H] = P) is flipped three times. Assume that the value of P does not change during the sequence of tosses. a. What is the probability that all three flips are heads? b. Find the probability that the second flip is heads given that the first flip is heads. c. Is the second flip independent of the first...
Problem 3. A fair coin is flipped until five heads are observed. Find the probability mass...
Problem 3. A fair coin is flipped until five heads are observed. Find the probability mass function and the expectation of the number of tails shown until then.
a. Suppose that a fair coin is tossed 15 times. If 10 heads are observed, determine...
a. Suppose that a fair coin is tossed 15 times. If 10 heads are observed, determine an expression / equation for the probability that 7 heads occurred in the first 9 tosses. b. Now, generalize your result from part a. Now suppose that a fair coin is to be tossed n times. If x heads are observed in the n tosses, derive an expression for the probability that there were y heads observed in the first m tosses. Note the...
3. For a nonnegative integer-valued random variable X show that i-0 4. A coin comes up heads with...
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...
A coin with probability p of heads is tossed until the first head occurs. It is...
A coin with probability p of heads is tossed until the first head occurs. It is then tossed again until the first tail occurs. Let X be the total number of tosses required. (i) Find the distribution function of X. (ii) Find the mean and variance of X
12. The total number of heads for a coin flipped four times is a random variable...
12. The total number of heads for a coin flipped four times is a random variable X with the following probability distribution P(X-0) 0.0625 PX-1) 0.2500 P(X-2) 0.3750 POX-3) 0.2500 P(X-4) 0.0625 Draw a graph of the density function. 13. The total number of heads for a coin flipped four times is a random variable X with the following probability distribution. P(X-0) 0.10 P(X-1) 0.40 P(X-2) 0.20 P(X-3) 0.10 P(X-4) 0.20 Determine the mean and variance of x.
6. A fair coin is flipped repeatedly until 50 heads are observed. What is the probability that at least 80 flips are necessary? (You may calculate an approximate answer.)
Assume that a coin is flipped where the probability of coin lands "Heads" is 0.49. The coin is flipped once more. This time, the probability of obtaining the first flip's result is 0.38. The random variable X is defined as the total number of heads observed in two flips. On the other hand, the random variable Y is defined as the absolute difference between the total number of heads and the total number of tails observed in two flips. Calculate...
Exercise 8.52. A fair coin is flipped 30 times. LetX denote the number of heads among the first 20 coin flips and Y denote the number of heads among the last 20 coin flips. Compute the correlation coefficient of X and I.
Coin with random bias. Let P be a random variable distributed uniformly over [0, 1]. A coin with (random) bias P (i.e., Pr[H] = P) is flipped three times. Assume that the value of P does not change during the sequence of tosses. a. What is the probability that all three flips are heads? b. Find the probability that the second flip is heads given that the first flip is heads. c. Is the second flip independent of the first...
Problem 3. A fair coin is flipped until five heads are observed. Find the probability mass function and the expectation of the number of tails shown until then.
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...
A coin with probability p of heads is tossed until the first head occurs. It is then tossed again until the first tail occurs. Let X be the total number of tosses required. (i) Find the distribution function of X. (ii) Find the mean and variance of X
12. The total number of heads for a coin flipped four times is a random variable X with the following probability distribution P(X-0) 0.0625 PX-1) 0.2500 P(X-2) 0.3750 POX-3) 0.2500 P(X-4) 0.0625 Draw a graph of the density function. 13. The total number of heads for a coin flipped four times is a random variable X with the following probability distribution. P(X-0) 0.10 P(X-1) 0.40 P(X-2) 0.20 P(X-3) 0.10 P(X-4) 0.20 Determine the mean and variance of x.
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