(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.

So, E[X] = Σ [x * P(X=x)] for all x

= (-30 * 0.45) + (-20 * 0.20) + (-10 * 0.10) + (0 * 0.10) + (10 * 0.05) + (20 * 0.10)
= -13.5 - 4 - 1 + 0 + 0.5 + 2
= -16

(b) The variance (Var[X]) of a random variable is calculated by finding the expectation of the squares of the variable (E[X^2]) and subtracting the square of the expectation of the variable (E[X])^2.

First, we find E[X^2] = Σ [x^2 * P(X=x)] for all x

= (-30^2 * 0.45) + (-20^2 * 0.20) + (-10^2 * 0.10) + (0^2 * 0.10) + (10^2 * 0.05) + (20^2 * 0.10)
= 405 + 80 + 10 + 0 + 5 + 40
= 540

Then, Var[X] = E[X^2] - (E[X])^2
= 540 - (-16)^2
= 540 - 256
= 284

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.

So, E[X] = Σ [x * P(X=x)] for all x

= (-30 * 0.45) + (-20 * 0.20) + (-10 * 0.10) + (0 * 0.10) + (10 * 0.05) + (20 * 0.10)
= -13.5 - 4 - 1 + 0 + 0.5 + 2
= -16

(b) The variance (Var[X]) of a random variable is calculated by finding the expectation of the squares of the variable (E[X^2]) and subtracting the square of the expectation of the variable (E[X])^2.

First, we find E[X^2] = Σ [x^2 * P(X=x)] for all x

= (-30^2 * 0.45) + (-20^2 * 0.20) + (-10^2 * 0.10) + (0^2 * 0.10) + (10^2 * 0.05) + (20^2 * 0.10)
= 405 + 80 + 10 + 0 + 5 + 40
= 540

Then, Var[X] = E[X^2] - (E[X])^2
= 540 - (-16)^2
= 540 - 256
= 284

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.

So, E[X] = Σ [x * P(X=x)] for all x

= (-30 * 0.45) + (-20 * 0.20) + (-10 * 0.10) + (0 * 0.10) + (10 * 0.05) + (20 * 0.10)
= -13.5 - 4 - 1 + 0 + 0.5 + 2
= -16

(b) The variance (Var[X]) of a random variable is calculated by finding the expectation of the squares of the variable (E[X^2]) and subtracting the square of the expectation of the variable (E[X])^2.

First, we find E[X^2] = Σ [x^2 * P(X=x)] for all x

= (-30^2 * 0.45) + (-20^2 * 0.20) + (-10^2 * 0.10) + (0^2 * 0.10) + (10^2 * 0.05) + (20^2 * 0.10)
= 405 + 80 + 10 + 0 + 5 + 40
= 540

Then, Var[X] = E[X^2] - (E[X])^2
= 540 - (-16)^2
= 540 - 256
= 284

Let X be a random variable with the following probability distribution.Value x of X P=Xx-30 0.45-20 0.20-10 0.100 0.1010 0.0520 0.10Complete 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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