Question

let \\(x\\) be a random variable with the following probability distribution.

value \\(x\\) of \\(x\\) | \\(p(x=x)\\)
10 | 0.25
20 | 0.40
30 | 0.10
40 | 0.20
50 | 0.05

complete the following. (if necessary, consult a list of formulas.)

(a) find the expectation \\(e(x)\\) of \\(x\\).
\\(e(x) = \\)

(b) find the variance \\(\text{var}(x)\\) of \\(x\\).
\\(\text{var}(x) = \\)

Explanation:

Calculate the expected value \(E(X)\)

Using the Discrete Random Variable and Probability Distribution knowledge points, we compute the expected value \(E(X)\) by summing the products of each value \(x\) and its corresponding probability \(P(X=x)\):

$$ E(X) = \sum x \cdot P(X=x) $$

$$ E(X) = (10 \cdot 0.25) + (20 \cdot 0.40) + (30 \cdot 0.10) + (40 \cdot 0.20) + (50 \cdot 0.05) $$

$$ E(X) = 2.5 + 8.0 + 3.0 + 8.0 + 2.5 = 24 $$

Calculate the expected value of \(X^2\)

To find the variance, we first compute \(E(X^2)\) using the probability distribution:

$$ E(X^2) = \sum x^2 \cdot P(X=x) $$

$$ E(X^2) = (10^2 \cdot 0.25) + (20^2 \cdot 0.40) + (30^2 \cdot 0.10) + (40^2 \cdot 0.20) + (50^2 \cdot 0.05) $$

$$ E(X^2) = (100 \cdot 0.25) + (400 \cdot 0.40) + (900 \cdot 0.10) + (1600 \cdot 0.20) + (2500 \cdot 0.05) $$

$$ E(X^2) = 25 + 160 + 90 + 320 + 125 = 720 $$

Calculate the variance \(\text{Var}(X)\)

We use the variance formula for a discrete random variable:

$$ \text{Var}(X) = E(X^2) - [E(X)]^2 $$

$$ \text{Var}(X) = 720 - 24^2 $$

$$ \text{Var}(X) = 720 - 576 = 144 $$

Answer:

Question 1

(a) Find the expectation \(E(X)\) of \(X\).
\(E(X) =\) <blank>24</blank>

Question 2

(b) Find the variance \(\text{Var}(X)\) of \(X\).
\(\text{Var}(X) =\) <blank>144</blank>