A fair coin is tossed three times. Let X denotes the number of TAILS observed. F

Question

A fair coin is tossed three times. Let X denotes the number of TAILS observed. Fill out the table below first to indicate the possible values of X and the corresponding probabilities.

x

f(x)

x*f(x)

(x-)2*f(x)

a.Use the definition of expectation and variance (in Chapter 4), and the table above to find: E(X), Var(X).

b.Is X a binomial random variable? Why or why not?

       If YES, list out the parameters: n, p.

c.If YES in b), use the formula for E(X) and Var(X) of a binomial random variable to calculate E(X) and Var(X). Compare with what you got in part a).

x

f(x)

x*f(x)

(x-)2*f(x)

Explanation / Answer

A fair coin is tossed three times.

The sample space is,

S = { HHH, HTH, HHT, HTT, THH, THT, TTH, TTT}

n(S) = 8

Let X denotes the number of TAILS observed.

Thepossible values of X are 0,1,2 and 3.

X=0 means 0 tail

X=1 means 1 tail

X=2 means 2 tails and

X=3 means 3 tails.

So the probability distribution of X is,

mean = x*f(x)

variance = (x - mean)2 * f(x)

Mean = 1.5

Variance = 0.75

Yes The distribution of X is binomial with parameters n=3 and p=1/2.

Because there are three possible values of X.

And probability of head = 1/2

and probability of tail = 1/2

q = 1 - p = 1 - 1/2 = 1/2

Mean and variance of binomial distribution is,

mean = n*p = 3*1/2 = 1.5

variance = n*p*q = 3*1/2*1/2 = 0.75

The mean and variances of Binomial distribution and by using a) are same.

x f(x) x*f(x) (x-mean)^2*f(x) 0 0.125 0 0.28125 1 0.375 0.375 0.09375 2 0.375 0.75 0.09375 3 0.125 0.375 0.28125 total 1 1.5 0.75

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