Question

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

value (x) of (x)	(p(x = x))
0	0.05
1	0.45
2	0.30
3	0.10
4	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 (var(x)) of (x).
(var(x)=)

Explanation:

Step1: Recall expectation formula

$E(X)=\sum_{i}x_{i}P(X = x_{i})$

Step2: Calculate expectation

$E(X)=(- 1)\times0.05+0\times0.05 + 1\times0.45+2\times0.30 + 3\times0.10+4\times0.05$
$E(X)=-0.05 + 0+0.45+0.60 + 0.30+0.20$
$E(X)=1.5$

Step3: Recall variance formula

$Var(X)=\sum_{i}(x_{i}-E(X))^{2}P(X = x_{i})$

Step4: Calculate variance

$Var(X)=(-1 - 1.5)^{2}\times0.05+(0 - 1.5)^{2}\times0.05+(1 - 1.5)^{2}\times0.45+(2 - 1.5)^{2}\times0.30+(3 - 1.5)^{2}\times0.10+(4 - 1.5)^{2}\times0.05$
$Var(X)=(-2.5)^{2}\times0.05+(-1.5)^{2}\times0.05+(-0.5)^{2}\times0.45+(0.5)^{2}\times0.30+(1.5)^{2}\times0.10+(2.5)^{2}\times0.05$
$Var(X)=6.25\times0.05 + 2.25\times0.05+0.25\times0.45+0.25\times0.30+2.25\times0.10+6.25\times0.05$
$Var(X)=0.3125+0.1125 + 0.1125+0.075+0.225+0.3125$
$Var(X)=1.15$

Answer:

(a) $E(X)=1.5$
(b) $Var(X)=1.15$