Let 𝑋 and 𝑌 be independent random variables, uniformly distributed on [0,1]. Let 𝑈=min{𝑋,𝑌} and 𝑉=max{𝑋,𝑌}. Let 𝑎=𝐄[𝑈𝑉] and 𝑏=𝐄[𝑉].
1. Find 𝑎. 2. Find 𝑏. 3. Find Cov(𝑈,𝑉).
You can give either a numerical answer or a symbolic expression involving 𝑎 and 𝑏. (There are many possible equivalent answers.)
1. 1/4
2. 1/2
3. 0
To find 𝑎 and 𝑏, we need to first find the joint probability density function (PDF) of 𝑋 and 𝑌, and then use it to calculate the expectations 𝐄[𝑈𝑉] and 𝐄[𝑉]. Finally, we can find the covariance Cov(𝑈,𝑉).
1. 𝐄[𝑈𝑉]:
To find 𝐄[𝑈𝑉], we need to find the joint PDF of 𝑈 and 𝑉, and then integrate their product over the appropriate range.
Since 𝑋 and 𝑌 are independent and uniformly distributed on [0,1], their joint PDF is given by:
𝑓(𝑋,𝑌) = 𝑓(𝑋) × 𝑓(𝑌) = 1 × 1 = 1, for 0 ≤ 𝑋 ≤ 1 and 0 ≤ 𝑌 ≤ 1.
Now we can express 𝑈 and 𝑉 in terms of 𝑋 and 𝑌:
𝑈 = min{𝑋,𝑌}, so 𝑈 = 𝑋 when 𝑋 ≤ 𝑌, and 𝑈 = 𝑌 when 𝑌 < 𝑋.
𝑉 = max{𝑋,𝑌}, so 𝑉 = 𝑋 when 𝑋 ≥ 𝑌, and 𝑉 = 𝑌 when 𝑌 ≥ 𝑋.
We need to consider these cases separately:
Case 1: When 𝑋 ≤ 𝑌
In this case, 𝑈 = 𝑋, 𝑉 = 𝑌, and their joint PDF is:
𝑓(𝑈,𝑉) = 𝑓(𝑋) × 𝑓(𝑌) = 1, for 0 ≤ 𝑋 ≤ 𝑌 ≤ 1.
Now let's calculate 𝐄[𝑈𝑉]:
𝐄[𝑈𝑉] = ∫∫(𝑈𝑉) × 𝑓(𝑈,𝑉)𝑑𝑈𝑑𝑉
= ∫∫(𝑋𝑌) × 1 𝑑𝑋𝑑𝑌
= ∫[0,1] ∫[𝑋,1] (𝑋𝑌) 𝑑𝑌𝑑𝑋
Integrating this double integral will give us the value of 𝑎.
2. 𝐄[𝑉]:
To find 𝐄[𝑉], we need to find the marginal PDF of 𝑉 and then calculate the expectation using this PDF.
The marginal PDF of 𝑉 can be found by integrating the joint PDF over the appropriate range. Since 𝑉 can take the maximum value from 𝑋 and 𝑌, we integrate over the entire range of 𝑋 and 𝑌:
𝑓(𝑉) = ∫∫𝑓(𝑋,𝑌)𝑑𝑋𝑑𝑌
= ∫[0,1]∫[0,1] 1 𝑑𝑋𝑑𝑌
Integrating this double integral will give us the marginal PDF of 𝑉. Then we can calculate 𝐄[𝑉] by taking the expectation using this PDF.
3. Cov(𝑈,𝑉):
The covariance Cov(𝑈,𝑉) can be calculated using the formula:
Cov(𝑈,𝑉) = 𝐄[𝑈𝑉] - 𝐄[𝑈] × 𝐄[𝑉]
We already know how to find 𝐄[𝑈𝑉], 𝐄[𝑈], and 𝐄[𝑉] from the steps above. By substituting these values into the covariance formula, we can calculate Cov(𝑈,𝑉).
Note: The calculations for 𝑎 and 𝑏 involve definite integrals, and it may be easier to evaluate them numerically using appropriate software or numerical methods.