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3.1 An experiment consists of tossing a fair coin 5 times. (a) Find the probability mass...

3.1 An experiment consists of tossing a fair coin 5 times. (a) Find the probability mass and distribution functions for the number of heads realized. (b) Find the probability of realizing heads at least 3 times out of the 5 trials.

3.1 An experiment consists of tossing a fair coin 5 times. (a) Find the probability mass and distribution functions for the number of heads realized. (b) Find the probability of realizing heads at least 3 times out of the 5 trials.
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a) here as number of trails are independent and probability of head on each event is fixed ; therefore it is binomial distribution with parameter n=5 and p=0.5

hence probability mass function of X:

P(X=x)=(0.5) (0.5)5-

frm above P(X=0)=\tiny \binom{5}{0}(0.5)^{0}(0.5)^{5} =0.03125

P(X=1)=\tiny \binom{5}{1}(0.5)^{1}(0.5)^{4} =0.15625

P(X=2)=0.3125

P(X=3)=0.3125

P(X=4)=0.15625

P(X=5)=0.03125

probbaility distribution function F(x) of X is as follows:

0 x<=0
0.03125 0<=x<1
0.1875 1<=x<2
F(x)= 0.5 2<=x<3
0.8125 3<=x<4
0.96875 4<=x<5
1 x>=5

b)

P(realizing at least 3 heads out of 5) =P(X=3)+P(X=4)+P(X=5) =0.3125+0.15625+0.03125=0.5

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