X is a random variable, such that X = 1 with probability 1/4, 2 with probability 3/4. What is the value of var(X^2)? What is the value of var(X^2 - 5X + 3)? What is the value of var((X - 6)^4)?
To find the variance of a random variable, we need to know its probability distribution. In this case, we have the following information about the random variable X:
X = 1 with probability 1/4
X = 2 with probability 3/4
Let's calculate the expected value of X^2 first:
E(X^2) = (1^2) * (1/4) + (2^2) * (3/4)
= 1/4 + 12/4
= 13/4
Next, we can calculate the variance of X^2 using the formula:
var(X^2) = E(X^2) - [E(X)]^2
= 13/4 - (3/4)^2
= 13/4 - 9/16
= 39/16 - 9/16
= 30/16
= 15/8
Now, let's calculate the variance of X^2 - 5X + 3:
First, we need to find the expected value of X^2 - 5X + 3:
E(X^2 - 5X + 3) = E(X^2) - 5 * E(X) + 3
= 13/4 - 5 * 3/4 + 3
= 13/4 - 15/4 + 3
= 13/4 - 15/4 + 12/4
= 10/4
= 5/2
Next, we can calculate the variance using the formula:
var(X^2 - 5X + 3) = E[(X^2 - 5X + 3)^2] - [E(X^2 - 5X + 3)]^2
Now, to calculate var((X - 6)^4), we will use the probability distribution of X:
X = 1 with probability 1/4
X = 2 with probability 3/4
Using this information, we can calculate the expected value of (X - 6)^4:
E((X - 6)^4) = (1 - 6)^4 * (1/4) + (2 - 6)^4 * (3/4)
= 5^4 * (1/4) + (-4)^4 * (3/4)
= 625/4 + 256 * 3/4
= 625/4 + 768/4
= 1393/4
Finally, we can calculate the variance of (X - 6)^4 using the formula:
var((X - 6)^4) = E[(X - 6)^4] - [E((X - 6)^4)]^2