Question

"We denote by X the profit (in USD bn) of Microsoft in the third quarter of 2023, and by Y the profit (in USD bn) of Apple in the third quarter of 2023. A financial analyst models the joint probability density function of X and Y by f(x,y) = (18x^2) if 0 ≤ x ≤ 3 and 0 ≤ y ≤ 2, otherwise.

(a) Show that f(x,y) is a joint probability density function.
(b) Compute the probability that Microsoft's profit is greater than 2 and Apple's profit is greater than 1.
(c) Compute the probability that Apple's profit is greater than 1."

Answer 1

To show that a function is a joint probability density function, it must be non-negative and its integral over the entire space must equal 1. Upon adjusting with a normalizing constant, the probability that Microsoft's profit is greater than 2 and Apple's profit is greater than 1 is 19/27. The probability that Apple's profit is greater than 1 is 1.

Step-by-step explanation:

Validating and Using Joint Probability Density Function

To show that f(x,y) is a joint probability density function (PDF), the function must satisfy two conditions:

1. The function must be non-negative for all x and y in the given domain.
2. The integral of f(x,y) over the entire space must equal 1, satisfying the property of a PDF that the total probability is 1.

Condition 1: In our case, since f(x,y) = 18x^2 and we are given that x is between 0 and 3, the function is always non-negative within the given domain.

Condition 2: To check the second condition, compute the double integral of f(x,y) over the specified limits.

∫∫ f(x, y) dy dx = ∫ from 0 to 3 (18x^2) dx ∫ from 0 to 2 dy
= (18 ∫ from 0 to 3 x^2 dx) (∫ from 0 to 2 dy)
= (18 [x^3 / 3] from 0 to 3) (y from 0 to 2)
= (18 [27 / 3 - 0]) (2 - 0)
= (18 [9]) (2)
= 162

Since this integral is 162, not 1, we must involve a normalizing constant C, where C = 1/162 to ensure the total probability is 1. Hence, the correct f(x,y) is (1/162)*18x^2 = (1/9)x^2, for 0 ≤ x ≤ 3 and 0 ≤ y ≤ 2.

Now to calculate the different probabilities:

(b) Probability that Microsoft's profit is greater than 2 and Apple's profit is greater than 1:

P(X>2, Y>1) = ∫ from 2 to 3 ∫ from 1 to 2 (1/9)x^2 dy dx
= ∫ from 2 to 3 (1/9)x^2 dx ∫ from 1 to 2 dy
= (1/9) [x^3 / 3] from 2 to 3 (y from 1 to 2)
= (1/9) [(27/3) - (8/3)] (1)
= (1/9) [19/3]
= 19/27

(c) Probability that Apple's profit is greater than 1:

P(Y>1) = ∫ from 0 to 3 ∫ from 1 to 2 (1/9)x^2 dy dx
= ∫ from 0 to 3 (1/9)x^2 dx ∫ from 1 to 2 dy
= (1/9) [x^3 / 3] from 0 to 3 (y from 1 to 2)
= (1/9) [(27/3) - 0] (1)
= (1/9) [9]
= 1

Therefore, the probability that Apple's profit is greater than 1 is 1. This implies that, given the range of Apple's profits is from 0 to 2, it is certain that Apple's profit will be greater than 1.