Question

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

value (x) of (x)	(p(x = x))
50	0.15
60	0.10
70	0.30
80	0.15
90	0.15

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

The formula for the expectation $E(X)$ of a discrete - random variable is $E(X)=\sum_{i}x_{i}P(X = x_{i})$.

$$E(X)=40\times0.15 + 50\times0.15+60\times0.10 + 70\times0.30+80\times0.15+90\times0.15$$

Step2: Calculate each product

$$40\times0.15 = 6$$

$$50\times0.15=7.5$$

$$60\times0.10 = 6$$

$$70\times0.30 = 21$$

$$80\times0.15=12$$

$$90\times0.15 = 13.5$$

Step3: Sum up the products

$$E(X)=6 + 7.5+6+21+12+13.5=66$$

Step4: Recall variance formula

The formula for the variance $Var(X)=\sum_{i}(x_{i}-E(X))^{2}P(X = x_{i})$. First, calculate $(x_{i}-E(X))^{2}$ for each $x_{i}$:
For $x = 40$: $(40 - 66)^{2}=(-26)^{2}=676$
For $x = 50$: $(50 - 66)^{2}=(-16)^{2}=256$
For $x = 60$: $(60 - 66)^{2}=(-6)^{2}=36$
For $x = 70$: $(70 - 66)^{2}=4^{2}=16$
For $x = 80$: $(80 - 66)^{2}=14^{2}=196$
For $x = 90$: $(90 - 66)^{2}=24^{2}=576$
Then $Var(X)=676\times0.15+256\times0.15 + 36\times0.10+16\times0.30+196\times0.15+576\times0.15$

$$Var(X)=(676 + 256+36+16+196+576)\times0.15$$

$$Var(X)=(676+256)+(36 + 16)+(196+576)\times0.15$$

$$Var(X)=932+52+772\times0.15$$

$$Var(X)=(932+52+772)\times0.15$$

$$Var(X)=1760\times0.15 = 264$$

Answer:

(a) $E(X)=66$
(b) $Var(X)=264$