Question

# A box contains 7 items, 4 of which are defective. a random sample of 3 items...

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.

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)=4Cx*3C3-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))

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