Question

Suppose a fair coin is tossed repeatedly and independently until the pair HT occurs (i.e., a...

Suppose a fair coin is tossed repeatedly and independently until the pair HT occurs (i.e., a head followed by a tail). Let X be the number of trials required to obtain the ordered pair HT.

  • (a) Show that the probability mass function of X is given by p(x) = (x − 1)/2x. (Hint: Draw a tree and note that at stage x, for x ≥ 2, there are precisely x − 1 paths in which an H was obtained at stage x − 1.)
  • (b) Show that the moment-generating function of X is m(t) = e2t/(et − 2)2.
Math   Statistics-And-Probability   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
A fair coin is tossed. If a head occurs 1 die is rolled, if a tail...
A fair coin is tossed. If a head occurs 1 die is rolled, if a tail occurs 2 dice are rolled. Let X be the total on the die or dice. What is E[X]? What is the probablity of the loss?
A fair coin is tossed. If a head occurs 1 die is rolled, if a tail...
A fair coin is tossed. If a head occurs 1 die is rolled, if a tail occurs 2 dice are rolled. Let X be the total on the die or dice. E[X] is 5.25. What is the probablity of the loss? loss for X
A coin is tossed repeatedly; on each toss, a head is shown with probability p or...
A coin is tossed repeatedly; on each toss, a head is shown with probability p or a tail with probability 1 − p. All tosses are independent. Let E denote the event that the first run of r successive heads occurs earlier than the first run of s successive tails. Let A denote the outcome of the first toss. Show that P(E|A=head)=pr−1 +(1−pr−1)P(E|A=tail). Find a similar expression for P (E | A = tail) and then find P (E).
Using R or R-studio. 3. A fair coin is tossed until the first head occurs. Do...
Using R or R-studio. 3. A fair coin is tossed until the first head occurs. Do this experiment T = 10; 100; 1,000; 10,000 times in R, and plot the relative frequencies of this occurring at the ith toss, for suitable values of i. Compare this plot to the pmf that should govern such an experiment. Show that they converge as T increases. What is the expected number of tosses required? For each value of T, what is the sample...
A fair coin is tossed for n times independently. (i) Suppose that n = 3. Given...
A fair coin is tossed for n times independently. (i) Suppose that n = 3. Given the appearance of successive heads, what is the conditional probability that successive tails never appear? (ii) Let X denote the probability that successive heads never appear. Find an explicit formula for X. (iii) Let Y denote the conditional probability that successive heads appear, given no successive heads are observed in the first n − 1 tosses. What is the limit of Y as n...
1. A fair coin is tossed ten times. (a) What is the probability that all ten...
1. A fair coin is tossed ten times. (a) What is the probability that all ten tosses produce the small result? (b) What is the probability that the results alternate, i,e., Tail is followed by Head and head is followed by Tail? (c) What is the probability that the first five tosses produce identical results? 2. A point M is chosen in random within the unit square . (a) What is the probability that M is closer to Y- axis...
Suppose you toss an unfair coin 8 times independently. The probability of getting heads is 0.3....
Suppose you toss an unfair coin 8 times independently. The probability of getting heads is 0.3. Denote the outcome to be 1 if you get heads and 0 if you get tails. 1.Write down the sample space. 2. What is the probability of the event that you get a head or a tail at least once? 3. If you get 8 same toss you will get x dollars, otherwise you will lose one dollar. On average, how large should x...
A biased coin (one that is not evenly balanced) is tossed 6 times. The probability of...
A biased coin (one that is not evenly balanced) is tossed 6 times. The probability of Heads on any toss is 0.3. Let X denote the number of Heads that come up. 1. Does this experiment meet the requirements to be considered a Bernoulli Trial? Explain why or why not. 2. If we call Heads a success, what would be the parameters of the binomial distribution of X? (Translation: find the values of n and p) 3. What is the...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT