A balanced coin is tossed three times. Let Y equal the

Solution 123E

Step1 of 5:

Let us consider a random variable Y it presents the number of heads observed.

Also we have n = 3.

Here our goal is:

a). We need to calculate the probabilities associated with Y = 0, 1, 2, and 3. By using Use the formula for the binomial probability distribution.

b). We need to construct a probability distribution similar to the one in Table 3.1.

c). We need to find the expected value and standard deviation of Y, using the formulas

E(Y ) = np and V (Y ) = npq.

d). We need to find the fraction of the population measurements lying within 1 standard deviation of the mean. Repeat for 2 standard deviations. And also compare with the results of Tchebysheff’s theorem and the empirical rule.

Step2 of 5:

a).

Probability of getting head is given by:

      =  

Here Y follows binomial distribution with parameter ‘n and p’.

That is YB(n, p)

                 YB(3, 0.5)

Then the probability mass function of binomial distribution is given by:

Where,

y = random variable

p = probability of success

n = sample size

Now,

For Y = 0 the probability is given  by:

Hence, P(Y = 0) = 0.1250.

For Y = 1 the probability is given  by:

Hence, P(Y = 1) = 0.3750.

For Y = 2 the probability is given  by: