To find the expected value (mean) [tex]\(\mu\)[/tex] of a random variable given its probability distribution, we use the following formula:
[tex]\[
\mu = \sum x_i P_i
\][/tex]
where [tex]\(x_i\)[/tex] represents the value of the random variable and [tex]\(P_i\)[/tex] represents the probability of [tex]\(x_i\)[/tex].
Given the values and their respective probabilities:
[tex]\[
\begin{array}{c|c}
x & P \\
\hline
0 & 0.08 \\
1 & 0.12 \\
2 & 0.22 \\
3 & 0.24 \\
4 & 0.18 \\
5 & 0.16 \\
\end{array}
\][/tex]
We calculate the expected value by multiplying each value of the random variable [tex]\(x_i\)[/tex] by its corresponding probability [tex]\(P_i\)[/tex] and then summing these products:
[tex]\[
\mu = (0 \times 0.08) + (1 \times 0.12) + (2 \times 0.22) + (3 \times 0.24) + (4 \times 0.18) + (5 \times 0.16)
\][/tex]
Let's compute each term individually:
[tex]\[
0 \times 0.08 = 0
\][/tex]
[tex]\[
1 \times 0.12 = 0.12
\][/tex]
[tex]\[
2 \times 0.22 = 0.44
\][/tex]
[tex]\[
3 \times 0.24 = 0.72
\][/tex]
[tex]\[
4 \times 0.18 = 0.72
\][/tex]
[tex]\[
5 \times 0.16 = 0.8
\][/tex]
Next, we sum these products:
[tex]\[
\mu = 0 + 0.12 + 0.44 + 0.72 + 0.72 + 0.8 = 2.8
\][/tex]
Therefore, the expected value [tex]\(\mu\)[/tex] is:
[tex]\[
\mu = 2.8
\][/tex]
