A student offered random people 5 cookies total Let X be the

A student offered random people 5 cookies total. Let X
be the number of cookies that the person took, and let
X have the following probability distribution (you may
assume that these are population values):
X 0 1 2 3 4 5
P(X) 0.05 0.25 0.38 0.18 0.11 k
(a) Find the value k such that X has a valid probability
distribution.
(b) Find Mx.
(c) Find variance
(d) What is the probability of three or more cookies be-
ing taken?
(e) What is the most likely number of cookies that a
random person will take?
(f) If you know that someone took less than 3 cookies,
what is the probability they took 1 cookie?

Solution

(a) 0.05+0.25+0.38+0.18+0.11+k=1

So k=0.03

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(b) E(X)= mean=0*0.05+1*0.25+2*0.38+3*0.18+4*0.11+5*0.03 =2.14

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(c) E(X^2)=0*0.05+1*0.25+2^2*0.38+3^2*0.18+4^2*0.11+5^2*0.03 = 5.9

Variance= E(X^2) - [E(X)]^2

=5.9 -2.14^2

=1.3204

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(d) P(X>=3) = P(X=3)+P(X=4)+P(X=5)

=0.18+0.11+0.03

=0.32

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(e)number of 2 cookies

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(f) P(X=1|X<3) = P(X=1 and X<3)/ P(X<3)

=0.25/(0.05+0.25+0.38)

=0.3676471