(a) The expectation or expected value (E[X]) of a random variable is calculated by summing the product of each outcome and its probability.

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

E[X] = Σ [x * P(X=x)]
= (-10 * 0.05) + (0 * 0.05) + (10 * 0.45) + (20 * 0.45)
= -0.5 + 0 + 4.5 + 9
= 13

So, the expectation E[X] of X is 13.

(b) The variance (Var[X]) of a random variable is calculated by summing the product of the square of the difference between each outcome and the expectation, and its probability.

For the given probability distribution, the variance Var[X] is calculated as follows:

Var[X] = Σ [(x - E[X])^2 * P(X=x)]
= [(-10 - 13)^2 * 0.05] + [(0 - 13)^2 * 0.05] + [(10 - 13)^2 * 0.45] + [(20 - 13)^2 * 0.45]
= [529 * 0.05] + [169 * 0.05] + [9 * 0.45] + [49 * 0.45]
= 26.45 + 8.45 + 4.05 + 22.05
= 61

So, the variance Var[X] of X is 61.

Question

(a) The expectation or expected value (E[X]) of a random variable is calculated by summing the product of each outcome and its probability.

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

E[X] = Σ [x * P(X=x)] 
     = (-10 * 0.05) + (0 * 0.05) + (10 * 0.45) + (20 * 0.45)
     = -0.5 + 0 + 4.5 + 9
     = 13

So, the expectation E[X] of X is 13.

(b) The variance (Var[X]) of a random variable is calculated by summing the product of the square of the difference between each outcome and the expectation, and its probability.

For the given probability distribution, the variance Var[X] is calculated as follows:

Var[X] = Σ [(x - E[X])^2 * P(X=x)]
       = [(-10 - 13)^2 * 0.05] + [(0 - 13)^2 * 0.05] + [(10 - 13)^2 * 0.45] + [(20 - 13)^2 * 0.45]
       = [529 * 0.05] + [169 * 0.05] + [9 * 0.45] + [49 * 0.45]
       = 26.45 + 8.45 + 4.05 + 22.05
       = 61

So, the variance Var[X] of X is 61.

Knowee AI · Accepted Answer

(a) The expectation or expected value (E[X]) of a random variable is calculated by summing the product of each outcome and its probability.

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

E[X] = Σ [x * P(X=x)] 
     = (-10 * 0.05) + (0 * 0.05) + (10 * 0.45) + (20 * 0.45)
     = -0.5 + 0 + 4.5 + 9
     = 13

So, the expectation E[X] of X is 13.

(b) The variance (Var[X]) of a random variable is calculated by summing the product of the square of the difference between each outcome and the expectation, and its probability.

For the given probability distribution, the variance Var[X] is calculated as follows:

Var[X] = Σ [(x - E[X])^2 * P(X=x)]
       = [(-10 - 13)^2 * 0.05] + [(0 - 13)^2 * 0.05] + [(10 - 13)^2 * 0.45] + [(20 - 13)^2 * 0.45]
       = [529 * 0.05] + [169 * 0.05] + [9 * 0.45] + [49 * 0.45]
       = 26.45 + 8.45 + 4.05 + 22.05
       = 61

So, the variance Var[X] of X is 61.

Let X be a random variable with the following probability distribution.Value x of X P=Xx-10 0.050 0.0510 0.4520 0.45Complete 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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