Question

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Question 2. Consider a variable x that describes the number of under-inflated tire on a four-wheel automobile. The mass function is given by: p(0) 0.4, p(1) p(2) p(3) 0.1; p(4) 0.3 a) Verify that this is a proper mass function. b) Determine the expected value and the variance of this distribution. mean value?

Answer 1

a)

here as probability of each data point is between 0 and 1 ; and sum of probability over sample space

=P(0)+P(1)+P(2)+P(3)+P(4)=0.4+0.1+0.1+0.1+0.3=1

hence this is valid mass function

b)

x	P(x)	xP(x)	x2P(x)
0	0.4	0.000	0.000
1	0.1	0.100	0.100
2	0.1	0.200	0.400
3	0.1	0.300	0.900
4	0.3	1.200	4.800
	total	1.800	6.200
	E(x) =μ=	ΣxP(x) =	1.8000
	E(x2) =	Σx2P(x) =	6.2000
	Var(x)=σ2 =	E(x2)-(E(x))2=	2.960

from above expected value E(X)=1.8

and variance =2.96

c)

as std deviation =sqrt(2.96)=1.7205

values -/+ 1 std deviation from mean =1.8-/+1.7205 =0.08 to 3.52

hence proportion of cars within -/+ 1 std deviation from mean=P(0.08 <X<3.52)=P(1)+P(2)+P(3)=0.1+0.1+0.1=0.3