Question

A box contains 7 items, 4 of which are defective. a random sample of 3 items are taken from the box. Let X be the number of defective items in the sample. 1.Find the probability mass function of X. 2.Find the mean and the variance of X.

Answer 1

here X follows hypergeometric distribution with parameter n=3 ; k=4 and N =7

probability mass function of X =P(X=x)=P(getting x defective from 4 and 3-x good items from 3)

P(X=x)=⁴C_x*³C_3-x/⁷C₃

from above formula

x	P(x)
0	0.0286
1	0.3429
2	0.5143
3	0.1143

2)

x	P(x)	xP(x)	x2P(x)
0	0.0286	0.000	0.000
1	0.3429	0.343	0.343
2	0.5143	1.029	2.057
3	0.1143	0.343	1.029
	total	1.714	3.429
	E(x) =μ=	ΣxP(x) =	1.7143
	E(x2) =	Σx2P(x) =	3.4286
	Var(x)=σ2 =	E(x2)-(E(x))2=	0.490

from above mean =1.7143 (this can directly be derived from formula nk/N =3*4/7=1.7143)

and variance =0.490 ( this can directly be derived from formula nk/N(1-k/N)*(N-n)/(N-1))