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. Use the definition of expectation and variance (in Chapter 4), and the table above to find: E(X), Var(X). Is X a binomial random variable? Why or why not? If YES, list out the parameters n, p. lf 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).

Explanation / Answer

a) per definition

E[X]= Sum of (x*f(x)) =1.5

V[X]=E[(X-Mu)^2]=Sum of (x-Mu)^2*f(x)) = 0.75

Computation:

Formula:

b)Yes, X is binomial random variable since it is discrete also it is the count of number of successes in n Bernoulli trials.

Parameters:

X~B(n=3,p=0.5)

c) for Binomial distribution

E[X]= np = 3*0.5=1.5

V[X]= npq = 3*0.5*(1-0.5) = 0.75

These values are exactly same as computed in section b.

X f(x) x*f(x) (x-Mu)^2 (x-Mu)^2 *f(x) 0 0.125 0 2.25 0.281 1 0.375 0.375 0.25 0.094 2 0.375 0.75 0.25 0.094 3 0.125 0.375 2.25 0.281 mean (Mu)= 1.5 variance= 0.75

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