Question

what is the expected value of the probability distribution of the discrete random variable x?
x p(x = x)
2 0.07
4 0.19
6 0.25
8 0.11
10 0.07
12 0.30
14 0.01
o μ = 71.52
o μ = 10.98
o μ = 11.21
o μ = 7.72

Explanation:

Step1: Recall expected - value formula

The expected value $\mu$ of a discrete random variable $X$ is given by $\mu=\sum_{i}x_{i}P(X = x_{i})$.

Step2: Calculate product for each pair

For $x_1 = 2$, $P(X = 2)=0.07$, product is $2\times0.07 = 0.14$.
For $x_2 = 4$, $P(X = 4)=0.19$, product is $4\times0.19 = 0.76$.
For $x_3 = 6$, $P(X = 6)=0.25$, product is $6\times0.25 = 1.5$.
For $x_4 = 8$, $P(X = 8)=0.11$, product is $8\times0.11 = 0.88$.
For $x_5 = 10$, $P(X = 10)=0.07$, product is $10\times0.07 = 0.7$.
For $x_6 = 12$, $P(X = 12)=0.30$, product is $12\times0.30 = 3.6$.
For $x_7 = 14$, $P(X = 14)=0.01$, product is $14\times0.01 = 0.14$.

Step3: Sum up the products

$\mu=0.14 + 0.76+1.5 + 0.88+0.7+3.6+0.14 = 7.72$.

Answer:

$\mu = 7.72$