Question

9. Let Y be a random variable who probability density function is given by P(Y- y) = 옮 when y is an intger between 1 and n inclusive. What should n be for this to be valid probability density function? Compute E(Y), V(Y), E(17Y - T) and 10, 10% of bottles produced at a factory have cracks. If two bottles are selected, find the mean and variance of the number of cracked bottles selected.

Answer 1

9) here as sum of probabilities of all sample space =1

hence (1/10+2/10+..+n/10)=1

n*(n+1)/20 =1

n*(n+1)=20

from above n=4

x	f(x)	xP(x)	x2P(x)
1	0.100	0.100	0.100
2	0.200	0.400	0.800
3	0.300	0.900	2.700
4	0.400	1.600	6.400
	total	3.000	10.000
	E(x) =μ=	ΣxP(x) =	3.0000
	E(x2) =	Σx2P(x) =	10.0000
	Var(x)=σ2 =	E(x2)-(E(x))2=	1.0000

E(Y)=3

Var(Y)=1.00

E(17Y-$\pi$)=17*E(Y)-$\pi$=17*3-$\pi$ =51-$\pi$

E(1/Y)=(1/1)*(1/10)+(1/2)*(2/10)+(1/3)*(3/10)+(1/4)*(4/10)=0.4

10)

here this is binomial distribution with parameter n=2 and p=0.1

mean =np=2*0.1=0.2

variance =np(1-p)=2*0.1*(1-0.1)=0.18