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A balanced coin is tossed three times. Let Y equal the
Chapter 3, Problem 173E(choose chapter or problem)
Problem 173E
A balanced coin is tossed three times. Let Y equal the number of heads observed.
a Use the formula for the binomial probability distribution to calculate the probabilities associated with Y = 0, 1, 2, and 3.
b Construct a probability distribution similar to the one in Table 3.1.
c Find the expected value and standard deviation of Y, using the formulas E(Y ) = np and V (Y ) = npq.
d Using the probability distribution from part (b), find the fraction of the population measurements lying within 1 standard deviation of the mean. Repeat for 2 standard deviations. How do your results compare with the results of Tchebysheff’s theorem and the empirical rule?
Questions & Answers
QUESTION:
Problem 173E
A balanced coin is tossed three times. Let Y equal the number of heads observed.
a Use the formula for the binomial probability distribution to calculate the probabilities associated with Y = 0, 1, 2, and 3.
b Construct a probability distribution similar to the one in Table 3.1.
c Find the expected value and standard deviation of Y, using the formulas E(Y ) = np and V (Y ) = npq.
d Using the probability distribution from part (b), find the fraction of the population measurements lying within 1 standard deviation of the mean. Repeat for 2 standard deviations. How do your results compare with the results of Tchebysheff’s theorem and the empirical rule?
ANSWER:
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: