Question

5. A number is selected at random from the digits 1 through 10 Let X equal the number of distinct divisors of the number selected. (Note: 1 is a divisor of all integers, and each nonzero integer is also a divisor of itself.) Write the probability distribution of X and find its mean and standard deviation

Answer 1

below is pmf of X:

P(X=1)=P(selecting one as it has only one divisor) =1/10=0.1

P(X=2)=P(selecing 2,3,5,7) =4/10=0.4

P(X=3)=P(selecting 4,9)=2/10=0.2

P(X=4)=P(selecting 6,8,10)=3/10=0.3

x	P(x)	xP(x)	x2P(x)
1	0.1000	0.100	0.100
2	0.4000	0.800	1.600
3	0.2000	0.600	1.800
4	0.3000	1.200	4.800
	total	2.700	8.300
	E(x) =μ=	ΣxP(x) =	2.7000
	E(x2) =	Σx2P(x) =	8.3000
	Var(x)=σ2 =	E(x2)-(E(x))2=	1.0100
	std deviation=	σ= √σ2 =	1.0050

mean of X =E(X)=2.70

and standard deviation of X =SD(X)=1.0050