Find the expected value of the random variable with the given probability distribution.

[tex]\[ \mu = [?] \][/tex]

Do not round your answer.

\begin{tabular}{r|c}
[tex]$x$[/tex] & [tex]$P(x)$[/tex] \\
\hline
-3 & 0.02 \\
-1 & 0.34 \\
1 & 0.41 \\
3 & 0.15 \\
5 & 0.04 \\
7 & 0.02 \\
\end{tabular}



Answer :

To find the expected value (mean) of a random variable given its probability distribution, you need to use the formula for the expected value of a discrete random variable. The expected value [tex]\( \mu \)[/tex] is calculated by summing the products of each value of the random variable and its corresponding probability:

[tex]\[ \mu = \sum_{i} x_i \cdot P(x_i) \][/tex]

Given the probability distribution:

[tex]\[ \begin{array}{r|c} x & P \\ \hline -3 & 0.02 \\ -1 & 0.34 \\ 1 & 0.41 \\ 3 & 0.15 \\ 5 & 0.04 \\ 7 & 0.02 \\ \end{array} \][/tex]

We need to multiply each [tex]\(x\)[/tex] value with its corresponding probability [tex]\(P\)[/tex] and then sum up all these products.

Calculations:

[tex]\[ -3 \cdot 0.02 = -0.06 \][/tex]

[tex]\[ -1 \cdot 0.34 = -0.34 \][/tex]

[tex]\[ 1 \cdot 0.41 = 0.41 \][/tex]

[tex]\[ 3 \cdot 0.15 = 0.45 \][/tex]

[tex]\[ 5 \cdot 0.04 = 0.20 \][/tex]

[tex]\[ 7 \cdot 0.02 = 0.14 \][/tex]

Next, sum these products:

[tex]\[ \begin{align*} \mu &= -0.06 + (-0.34) + 0.41 + 0.45 + 0.20 + 0.14 \\ &= -0.06 - 0.34 + 0.41 + 0.45 + 0.20 + 0.14 \end{align*} \][/tex]

Adding these together:

[tex]\[ -0.06 - 0.34 + 0.41 + 0.45 + 0.20 + 0.14 = 0.80 \][/tex]

The expected value [tex]\( \mu \)[/tex] of the random variable is:

[tex]\[ \mu = 0.7999999999999999 \][/tex]

So, [tex]\(\boxed{0.7999999999999999}\)[/tex]