Question

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

value ( x ) of ( x )	( p(x = x) )
3	0.10
4	0.45
5	0.10

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

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

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

Explanation:

Step1: Calculate expectation $E(X)$

Multiply each $x$ by $P(X=x)$, sum results.
$$E(X) = (2 \times 0.35) + (3 \times 0.10) + (4 \times 0.45) + (5 \times 0.10)$$
$$E(X) = 0.7 + 0.3 + 1.8 + 0.5 = 3.3$$

Step2: Calculate $E(X^2)$

Square each $x$, multiply by $P(X=x)$, sum.
$$E(X^2) = (2^2 \times 0.35) + (3^2 \times 0.10) + (4^2 \times 0.45) + (5^2 \times 0.10)$$
$$E(X^2) = (4 \times 0.35) + (9 \times 0.10) + (16 \times 0.45) + (25 \times 0.10)$$
$$E(X^2) = 1.4 + 0.9 + 7.2 + 2.5 = 12.0$$

Step3: Calculate variance $Var(X)$

Use formula $Var(X) = E(X^2) - [E(X)]^2$.
$$Var(X) = 12.0 - (3.3)^2 = 12.0 - 10.89 = 1.11$$

Answer:

(a) $E(X) = 3.3$
(b) $Var(X) = 1.11$