Let X be the number of packages being mailed by a random customer at a certain s

Question

Let X be the number of packages being mailed by a random customer at a certain shipping facility, and that customers only send between one and four packages at a time, 25% of all customers send one package, 40% send two packages, 25% send three packages, and the rest send four packages. Consider a random sample of two customers and let X be the sample mean number of packages shipped. a) Obtain the probability distribution of X. b) Use your answer in part a to calculate P(X lessthanorequalto 3). c) What is the expected value of X? d) What is the variance of X?

Explanation / Answer

here let x1 and x2 are number of packages send by both the customers.

a) aditing duplicate value of Xbar; below is probabilty distribution:

b) P(X<=3)=0.0625+0.2+0.285+0.25+0.1425=0.94

c)E(X) =2.2

d)Var(X) =E(X2) -(E(X))2 =5.27-2.22 =0.43

x1 x2 p1 p2 Xbar P(x1,x2)=P1*P2 1 1 0.25 0.25 1 0.0625 2 1 0.4 0.25 1.5 0.1 3 1 0.25 0.25 2 0.0625 4 1 0.1 0.25 2.5 0.025 1 2 0.25 0.4 1.5 0.1 2 2 0.4 0.4 2 0.16 3 2 0.25 0.4 2.5 0.1 4 2 0.1 0.4 3 0.04 1 3 0.25 0.25 2 0.0625 2 3 0.4 0.25 2.5 0.1 3 3 0.25 0.25 3 0.0625 4 3 0.1 0.25 3.5 0.025 1 4 0.25 0.1 2.5 0.025 2 4 0.4 0.1 3 0.04 3 4 0.25 0.1 3.5 0.025 4 4 0.1 0.1 4 0.01

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