To find the expected value of a discrete random variable, we use the formula:
[tex]\[
E(X) = \sum (x \cdot P(x))
\][/tex]
In this case, we have the values of [tex]\( X \)[/tex] and their corresponding probabilities [tex]\( P(X) \)[/tex]:
[tex]\[
X = [-10, 8]
\][/tex]
[tex]\[
P(X) = [0.4, 0.6]
\][/tex]
We will calculate the expected value step by step.
1. Multiply each value of [tex]\( X \)[/tex] by its corresponding probability:
- For [tex]\( X = -10 \)[/tex] with [tex]\( P(X) = 0.4 \)[/tex]:
[tex]\[
-10 \times 0.4 = -4
\][/tex]
- For [tex]\( X = 8 \)[/tex] with [tex]\( P(X) = 0.6 \)[/tex]:
[tex]\[
8 \times 0.6 = 4.8
\][/tex]
2. Add these results together to find the expected value:
[tex]\[
-4 + 4.8 = 0.7999999999999998
\][/tex]
Thus, the expected value [tex]\( E(X) \)[/tex] of the given random variable is:
[tex]\[
0.7999999999999998
\][/tex]
