i need help with this A shipment contains 20 assembled circu

i need help with this

A shipment contains 20 assembled circuit boards of which five are defective. Ten circuit boards from the shipment are selected without replacement. Suppose that X denotes the number of defective boards out of the 10 selected. Find the probability distribution of the random variable X and then find its mean and variance.

Solution

total number of circuits=N=20

sample size=n=10 drawn without replacement.

number of defectives=5

X denotes the number of defectives in the sample

so X~Hypergeometric(20,5,10)

so the distribution of X is

X   p(X)
0   0.016254
1   0.135449
2   0.348297
3   0.348297
4   0.135449
5   0.016254

mean=1*0.135449+2*0.348297+3*0.348297+4*0.135449+5*0.016254=2.5

variance=1²*0.135449+2²*0.348297+3²*0.348297+4²*0.135449+5²*0.016254-2.5²=0.986844