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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) = (18/1)(x^2 * y^0), if 0 <= x <= 3 and 0 <= y <= 2, otherwise.

(a) Show that f(x, y) is a joint probability density function.

(b) Calculate the probability that Microsoft's profit is greater than 2 and Apple's profit is greater than 1.

(c) Calculate the probability that Apple's profit is greater than 1.

User Cem Polat
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Final answer:

a. The given function f(x, y) does not satisfy the requirements to be a joint probability density function (PDF). b. and c. Since the joint PDF is not valid, we cannot calculate the probabilities requested.

Step-by-step explanation:

a. To show that f(x, y) is a joint probability density function (PDF), we need to check two conditions:

1. The function must be non-negative for all (x, y) in its domain.
2. The integral of f(x, y) over its entire domain must be equal to 1.

For the given function f(x, y) = (18/1)(x^2 * y^0), the domain is defined as 0 <= x <= 3 and 0 <= y <= 2. Since the function is defined as a constant multiple of x^2 * y^0, which is non-negative for all (x, y) in its domain, condition 1 is satisfied. To check condition 2, we need to find the integral of f(x, y) over its domain:

∫∫ f(x, y) dx dy = ∫∫ (18/1)(x^2 * y^0) dx dy
= (18/1) ∫∫ (x^2 * y^0) dx dy
= (18/1) ∫0^3 ∫0^2 (x^2 * y^0) dx dy
= (18/1) ∫0^3 (x^2 * y^0)|0^2 dy
= (18/1) ∫0^3 (2^0 - 0^0) dy
= (18/1) ∫0^3 (1 - 1) dy
= (18/1) ∫0^3 0 dy
= 0

Since the integral is equal to 0 and not 1, this means that f(x, y) is not a valid joint PDF. Thus, the given function does not satisfy the requirements to be a joint probability density function.



b. The probability that Microsoft's profit is greater than 2 and Apple's profit is greater than 1 can be calculated by finding the area under the joint PDF, f(x, y), over the region where X > 2 and Y > 1. Since f(x, y) is not a valid joint PDF, we cannot calculate this probability.



c. The probability that Apple's profit is greater than 1 can be calculated by finding the area under the joint PDF, f(x, y), over the region where Y > 1. Since f(x, y) is not a valid joint PDF, we cannot calculate this probability.

User Nick De Jaeger
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