Question

Let (𝑋,𝑌) be a pair of random variables with joint density ℎ(𝑥,𝑦)=𝑥+𝑦 over the space [0,1]2.

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the marginal density functions for 𝑋 and 𝑌, we need to integrate the joint density ℎ(𝑥,𝑦) over the appropriate ranges.

The marginal density function of 𝑋 is found by integrating ℎ(𝑥,𝑦) with respect to 𝑦 over the entire range of 𝑦, which in this case is [0,1]:

𝑓𝑋(𝑥) = ∫[0,1] (𝑥+𝑦) d𝑦

Simplifying the integral, we have:

𝑓𝑋(𝑥) = 𝑥∫[0,1] d𝑦 + ∫[0,1] 𝑦 d𝑦
= 𝑥[y]₀¹ + [y²/2]₀¹
= 𝑥(1-0) + (1/2-0)
= 𝑥 + 1/2

Therefore, the marginal density function of 𝑋 is 𝑓𝑋(𝑥) = 𝑥 + 1/2.

Similarly, the marginal density function of 𝑌 is found by integrating ℎ(𝑥,𝑦) with respect to 𝑥 over the entire range of 𝑥, which is also [0, 1]:

𝑓𝑌(𝑦) = ∫[0,1] (𝑥+𝑦) d𝑥

Again, simplifying the integral, we have:

𝑓𝑌(𝑦) = ∫[0,1] 𝑥 d𝑥 + 𝑦∫[0,1] d𝑥
= [𝑥²/2]₀¹ + 𝑦[x]₀¹
= (1/2-0) + 𝑦(1-0)
= 1/2 + 𝑦

Therefore, the marginal density function of 𝑌 is 𝑓𝑌(𝑦) = 1/2 + 𝑦.

Answer 2

[AI-generated answer — Step-by-Step Bot]

To find the marginal density functions for 𝑋 and 𝑌, we need to integrate the joint density function ℎ(𝑥,𝑦) over the appropriate ranges.

For 𝑋:
To find the marginal density function for 𝑋, we integrate the joint density function ℎ(𝑥,𝑦) with respect to 𝑦, over the range [0,1].

∫(0 to 1) (𝑥 + 𝑦) d𝑦

Using the definite integral formula, we can solve this integral:

= 𝑥𝑦 + (1/2)𝑦^2 | (0 to 1)
= 𝑥 + (1/2)

So, the marginal density function for 𝑋 is ℎ𝑋(𝑥) = 𝑥 + (1/2).

For 𝑌:
To find the marginal density function for 𝑌, we integrate the joint density function ℎ(𝑥,𝑦) with respect to 𝑥, over the range [0,1].

∫(0 to 1) (𝑥 + 𝑦) d𝑥

= (1/2)𝑥^2 + 𝑥𝑦 | (0 to 1)
= (1/2) + 𝑦

So, the marginal density function for 𝑌 is ℎ𝑌(𝑦) = (1/2) + 𝑦.

Therefore, the marginal density function for 𝑋 is ℎ𝑋(𝑥) = 𝑥 + (1/2), and the marginal density function for 𝑌 is ℎ𝑌(𝑦) = (1/2) + 𝑦.