(a) The expectation or expected value (E[X]) of a random variable is calculated by multiplying each possible outcome by their respective probabilities and summing these products.

For the given probability distribution, the expectation E[X] is calculated as follows:

E[X] = (-20 * 0.05) + (-10 * 0.10) + (0 * 0.20) + (10 * 0.20) + (20 * 0.15) + (30 * 0.30)
= -1 + -1 + 0 + 2 + 3 + 9
= 12

(b) The variance (Var[X]) of a random variable is calculated by subtracting the square of the expectation from the expectation of the squares.

First, calculate E[X^2] (the expectation of the squares):

E[X^2] = (-20^2 * 0.05) + (-10^2 * 0.10) + (0^2 * 0.20) + (10^2 * 0.20) + (20^2 * 0.15) + (30^2 * 0.30)
= 200 + 100 + 0 + 200 + 600 + 2700
= 3800

Then, calculate Var[X] = E[X^2] - (E[X])^2:

Var[X] = 3800 - (12)^2
= 3800 - 144
= 3656

Question

(a) The expectation or expected value (E[X]) of a random variable is calculated by multiplying each possible outcome by their respective probabilities and summing these products.

For the given probability distribution, the expectation E[X] is calculated as follows:

E[X] = (-20 * 0.05) + (-10 * 0.10) + (0 * 0.20) + (10 * 0.20) + (20 * 0.15) + (30 * 0.30)
     = -1 + -1 + 0 + 2 + 3 + 9
     = 12

(b) The variance (Var[X]) of a random variable is calculated by subtracting the square of the expectation from the expectation of the squares.

First, calculate E[X^2] (the expectation of the squares):

E[X^2] = (-20^2 * 0.05) + (-10^2 * 0.10) + (0^2 * 0.20) + (10^2 * 0.20) + (20^2 * 0.15) + (30^2 * 0.30)
       = 200 + 100 + 0 + 200 + 600 + 2700
       = 3800

Then, calculate Var[X] = E[X^2] - (E[X])^2:

Var[X] = 3800 - (12)^2
       = 3800 - 144
       = 3656

Knowee AI · Accepted Answer

(a) The expectation or expected value (E[X]) of a random variable is calculated by multiplying each possible outcome by their respective probabilities and summing these products.

For the given probability distribution, the expectation E[X] is calculated as follows:

E[X] = (-20 * 0.05) + (-10 * 0.10) + (0 * 0.20) + (10 * 0.20) + (20 * 0.15) + (30 * 0.30)
     = -1 + -1 + 0 + 2 + 3 + 9
     = 12

(b) The variance (Var[X]) of a random variable is calculated by subtracting the square of the expectation from the expectation of the squares.

First, calculate E[X^2] (the expectation of the squares):

E[X^2] = (-20^2 * 0.05) + (-10^2 * 0.10) + (0^2 * 0.20) + (10^2 * 0.20) + (20^2 * 0.15) + (30^2 * 0.30)
       = 200 + 100 + 0 + 200 + 600 + 2700
       = 3800

Then, calculate Var[X] = E[X^2] - (E[X])^2:

Var[X] = 3800 - (12)^2
       = 3800 - 144
       = 3656

Let X be a random variable with the following probability distribution.Value x of X P=Xx-20 0.05-10 0.100 0.2010 0.2020 0.1530 0.30Complete the following. (If necessary, consult a list of formulas.)(a) Find the expectation EX of X.=EX (b) Find the variance VarX of X.=VarX

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