In the last example, we saw that the conditional distribution of 𝑋, which was a uniform over a smaller range (and in some sense, less uncertain), had a smaller variance, i.e., 𝖵𝖺𝗋(𝑋∣𝐴)≤𝖵𝖺𝗋(𝑋). Here is an example where this is not true. Let 𝑌 be uniform on {0,1,2} and let 𝐵 be the event that 𝑌 belongs to {0,2}.

Question

In the last example, we saw that the conditional distribution of 𝑋, which was a uniform over a smaller range (and in some sense, less uncertain), had a smaller variance, i.e., 𝖵𝖺𝗋(𝑋∣𝐴)≤𝖵𝖺𝗋(𝑋). Here is an example where this is not true. Let 𝑌 be uniform on {0,1,2} and let 𝐵 be the event that 𝑌 belongs to {0,2}.

a) What is the variance of 𝑌?

𝖵𝖺𝗋(𝑌)=

Answer 1

(a) 2/3

(b) 1

Answer 2

Well, well, well! It seems we have a variance question here. Let's see if we can clown our way through it.

Now, we have a uniform distribution 𝑌 that takes values from {0, 1, 2}. To find the variance of 𝑌, we need to know a couple of things.

First, we need to find the mean of 𝑌. Since 𝑌 is uniform, with equal probability for each value, the mean is simply the average of the possible values. In this case, the mean is (0 + 1 + 2)/3 = 1.

Next, to find the variance, we need to calculate the squared difference between each value and the mean, and then take the average of those squared differences. The formula for variance is a bit of a clown fest, but bear with me.

For each value 𝑦, we calculate (𝑦 - 𝑚𝑒𝑎𝑛)^2. Let's do this for each value in 𝑌:

(0 - 1)^2 = 1
(1 - 1)^2 = 0
(2 - 1)^2 = 1

To find the variance, we take the average of these squared differences. Summing them up, we get 1 + 0 + 1 = 2. Dividing by 3 (the number of values), we get 2/3.

So, the variance of 𝑌 is 𝖵𝖺𝗋(𝑌) = 2/3. And that's the end of our clown show!

Answer 3

To calculate the variance of 𝑌, we first need to determine the mean of 𝑌.

Since 𝑌 is uniformly distributed on {0, 1, 2}, each value has an equal probability of occurring. Therefore, the mean of 𝑌 is given by:

𝐸(𝑌) = (0 + 1 + 2)/3 = 3/3 = 1

Next, we can calculate the variance using the formula:

𝖵𝖺𝗋(𝑌) = 𝐸[(𝑌 - 𝐸(𝑌))^2]

Substituting the mean value into the formula, we get:

𝖵𝖺𝗋(𝑌) = 𝐸[(𝑌 - 1)^2]

Now, we'll compute the variance by considering each value of 𝑌 and its respective probability:

𝖵𝖺𝗋(𝑌) = [(0 - 1)^2 * 1/3] + [(1 - 1)^2 * 1/3] + [(2 - 1)^2 * 1/3]
= [(-1)^2 * 1/3] + [0^2 * 1/3] + [1^2 * 1/3]
= [1/3] + [0] + [1/3]
= 2/3

Therefore, the variance of 𝑌 is 2/3.

Answer 4

To calculate the variance of 𝑌, we need to know the values of 𝑌 and their corresponding probabilities. In this case, 𝑌 is uniform on the set {0, 1, 2}, which means that each value in the set is equally likely to occur.

To find the variance, we can use the formula:

𝖵𝖺𝗋(𝑌) = 𝑬[(𝑌 - 𝑬[𝑌])^2]

where 𝑬[𝑌] is the expected value of 𝑌.

Since 𝑌 is a uniform random variable, the expected value 𝑬[𝑌] can be calculated as the average of the possible values:

𝑬[𝑌] = (0 + 1 + 2) / 3 = 1

Now, let's calculate the variance:

𝖵𝖺𝗋(𝑌) = 𝑬[(𝑌 - 𝑬[𝑌])^2]

Since 𝑌 can take on values 0, 1, and 2, we'll substitute these values in the formula:

𝖵𝖺𝗋(𝑌) = (0 - 1)^2 * P(𝑌 = 0) + (1 - 1)^2 * P(𝑌 = 1) + (2 - 1)^2 * P(𝑌 = 2)

Since 𝑌 is uniform, each value has an equal probability of occurring, which is 1/3. Substituting these values, we get:

𝖵𝖺𝗋(𝑌) = (0 - 1)^2 * (1/3) + (1 - 1)^2 * (1/3) + (2 - 1)^2 * (1/3)

Simplifying the equation:

𝖵𝖺𝗋(𝑌) = (1)^2 * (1/3) + (0)^2 * (1/3) + (1)^2 * (1/3)

𝖵𝖺𝗋(𝑌) = 1/3 + 0 + 1/3

𝖵𝖺𝗋(𝑌) = 2/3

Therefore, the variance of 𝑌 is 2/3.