Question

The payoff X of a lottery ticket in the Tri-State Pick 3 game is $500 with probability 1/1000 and $0 the rest of the time. Assume the payoffs X and Y are for separate days and are independent from each other.

1. What price should Tri-State charge for a lottery ticket so that they can break even in the long run (average profit =$ 0).

b. Find the mean and standard deviation of the total payoff X+Y.

Answer 1

Answer:

Given,

let us assume that x be the payoff of a lottery ticket

x = 500 with probability of 1/1000

0 else

Here it is clearly given that x & y are independent payoffs

i.e.,

E(x + y) = E(x) + E(y)

E(x) = 500*1/1000 + 0

= 1/2

= 0.5

E(x) = 0.5

E(y) = 0.5

Now let us consider,

E(x + y) = E(x) + E(y)

= 0.5 + 0.5

E(x + y) = 1

Now variance var(x + y) = v(x) + v(y)

[since cov(x,y) = 0 due to x&y are independent]

we know that,

V(x) = E(x^2) - E(x)^2

E(x^2) = 500^2 * 1/1000 + 0

= 250000*1000

= 250

substitute in V(x)

V(x) = 250 - (0.5)^2

= 250 - 0.25

= 249.75

V(x) = V(y) = 249.75

Now V(x + y) = V(x) + V(y)

substitute values

V(x + y) = 249.75 + 249.75

= 499.5

Variance = 499.5

Standard deviation = sqrt(499.5)

= 22.349

Standard deviation = 22.35