Question

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.

Answer 1

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)=

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