Question

A box contains two tickets, one marked “+2” and the other one “+4”. One hundred random draws are made with replacement, and the sum of the draws is computed.

(a) What is the EV of the sum?

(b) What is the SE of the sum?

(c) The sum of the draws will be around _______, give or take _______ or so.

Answer 1

Let X be the ticket drawn from a random draw.

Then

X = +2 with P(X=2) = 0.5

X = +4 with P(X=4) = 0.5

Now

E(X) =

= 2 * 0.5 + 4 * 0.5

= 3

E(X²) =  

= 2² * 0.5 + 4² * 0.5

= 10

Var(X) = E(X²) - (E(X))² = 10 - 3² = 1

a) For 100 draws with replacement, let the sum be denoted by Y

Then Y =100X

and E(Y)

= E(100X)

= 100 * E(X)

= 100 * 3

= 300

This is the EV of the sum

b)

Var(Y)

= Var(100X)

= (100)² * Var(X)

= (100)² * 1

=(100)²

Therefore SE(Y) = (Var(Y))^0.5 = 100

This is the SE of the sum

c)

The sum of the draws will be around 300, give or take 100 or so.

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