Question

A start-up company has 2000 investors, that company loses investors at a rate of 10 per year. Every time the company loses an investor, the company gets a loss of $200,000. For every investor that remains the company makes a profit of $2,000. Let F be the total earnings the company makes in a year, and X be the number of investors the company loses.

1)Write a function that calculates yearly earnings F as a function of X

2)Find P(F < 0), the probability that earnings are negative

3)E[F]

4)What is the probability that the company loses exactly 5 investors in a given year, given that they have not lost any investors in the first half of the year

Answer 1

1)

Number of investors lost in a year = X

Number of investors remaining after a year = 2000 - X

Yearly earnings F = (2000 - X) * 2000 - 200000 X

= 4000000 - 2000X - 200000 X

= 4000000 - 202000X

2)

F < 0

4000000 - 202000X < 0

X = 4000000 / 202000 =  19.80 20

Assuming X ~ Poisson( = 10)

Using Normal approximation to Poisson distribution, X ~ Normal( = 10, = 10)

P(F < 0) = P(X 20)

= P(X > 19.5) (Using Continuity correction)

= P[Z > (19.5 - 10)/]

= P[Z > 3.00]

= 0.00135

3)

E[F] = E[4000000 - 202000X]

= 4000000 - 202000 E[X]

= 4000000 - 202000 * 10

= $1980000

4)

For half year, Y ~ Poisson( = 10 * 0.5 = 5)

Probability that the company loses exactly 5 investors in a given year, given that they have not lost any investors in the first half of the year

= Probability that the company loses exactly 5 investors in a second half year

= P(Y = 5)

= 0.1755