An experiment consists of the following A biased coin with a

An experiment consists of the following: A biased coin (with a probability of Heads equaling 0.75) is repeatedly tossed until a total of two tails are recorded. As examples, possible recorded sequences include: HTT: HHTHT; THHHT;.... Describe the sample space and the outcomes. A random variable X maps an outcome to the number of Heads in that outcome. Is this random variable one-to-one? Determine the range of this random variable and its PMF. A random variable Y is an indicator function of the presence of at least one Head in the outcome. In other words, it returns \'1\' if there is at least one Head in the sequence, and \'0\' otherwise. Determine the PMF of this random variable Y. Describe the events (subsets of sample space) given by {X > 3}, {X + 1 = 5}, {Y = 0}, {X = 3} {Y = 0} and determine their probabilities.

Solution

2) X = number of heads in an outcome.

note we stop the experiment when we will get second head. So we have x+2 trials.

No this random variable is not one to one as for example there are two outcomes as HTHHT and HHHTT, so both the casse X value is 3. so its many-one function.

Now PMF is given by f(x) = p^x q^2 where x = 0 to infinity

p = probability of coming head and q = 1-p

3) Y = 1 if atleast one head and 0 otherwise

Now PMF of Y is f(y) = 1-q^2 = 1- 0.0625 = 0.9375 when y=1

f(y) = q^2 = 0.0625 when y = 0

4) {X>=3} event of getting more than 3 heads. P(X>=3) = 1- P(X<=2) = 1- [{p * q^2} + {p^2 * q^2}] = 1- 0.082 = 0.918

P(X+1=5) =P(X=4) = 0.75^4 * 0.25^2

P(Y=0) = P(no head) = q^2

P(X=3 and Y=0) = 0 as {X=3) and {Y=0) = null set