To find the expected value of the given random variable, we will apply the formula for the expected value (or mean) of a discrete random variable. The expected value [tex]\( E(X) \)[/tex] is calculated as follows:
[tex]\[ E(X) = \sum_{i} (p_i \times x_i) \][/tex]
where [tex]\( p_i \)[/tex] is the probability of the [tex]\(i\)[/tex]-th outcome and [tex]\( x_i \)[/tex] is the value of that outcome.
Let's break down the calculations step by step using the provided probabilities and scores:
1. Identify the probabilities and corresponding scores:
[tex]\[
\begin{aligned}
&p_1 = 0.12, && x_1 = 0, \\
&p_2 = 0.13, && x_2 = 1, \\
&p_3 = 0.06, && x_3 = 2, \\
&p_4 = 0.13, && x_4 = 3, \\
&p_5 = 0.22, && x_5 = 9, \\
&p_6 = 0.34, && x_6 = 11.
\end{aligned}
\][/tex]
2. Calculate the products of each [tex]\( p_i \)[/tex] and [tex]\( x_i \)[/tex]:
[tex]\[
\begin{aligned}
&p_1 \times x_1 = 0.12 \times 0 = 0, \\
&p_2 \times x_2 = 0.13 \times 1 = 0.13, \\
&p_3 \times x_3 = 0.06 \times 2 = 0.12, \\
&p_4 \times x_4 = 0.13 \times 3 = 0.39, \\
&p_5 \times x_5 = 0.22 \times 9 = 1.98, \\
&p_6 \times x_6 = 0.34 \times 11 = 3.74.
\end{aligned}
\][/tex]
3. Sum these products to find the expected value:
[tex]\[
E(X) = 0 + 0.13 + 0.12 + 0.39 + 1.98 + 3.74 = 6.36.
\][/tex]
Therefore, the expected value of the random variable is [tex]\( 6.36 \)[/tex].
