Question

6. A statistics instructor has put 4 copies of his lecture notes in the library reserve section.
Let X represent the number of students who come in to borrow these notes. Suppose X
has a Poisson distribution with parameter lambda = 3. Find the expected number of copies
that have been borrowed. (Note: No more than 4 copies in total can be borrowed.)

I know the answer, and it is not 3. Please show me the explanation.

Answer 1

here below is probability distribution of X:

P(X=0)=P(no studetn came and threfore none is borrowed)=e^-3*3⁰/0! =0.0498

P(X=1)=P(1 student come )=e^-3*3¹/1! =0.1494

P(X=2)=e^-3*3²/2! =0.2240

P(X=3)=e^-3*3³/3! =0.2240

P(X=4)=P(4 or more students came)=1-0.0498-0.1494-0.2240-0.2240=0.3528

hence expected number of copies
that have been borrowed =E(X)=xP(x) =0*0.0498+1*0.1494+2*0.2240+3*0.2240+4*0.3528 = 2.6806