To find the expected value of the distribution, we follow these steps:
1. Define the random variable and probabilities: Let [tex]\( X \)[/tex] represent the number of tickets bought in one purchase. We have the following probability distribution:
- Number of tickets, [tex]\( X \)[/tex]: 1, 2, 3, 4, 5
- Corresponding probabilities, [tex]\( P(X) \)[/tex]: 0.29, 0.44, 0.19, 0.06, 0.02
2. Calculate the expected value: The expected value [tex]\( E(X) \)[/tex] of a discrete random variable [tex]\( X \)[/tex] is calculated using the formula:
[tex]\[
E(X) = \sum_{i} (x_i \cdot P(x_i))
\][/tex]
where [tex]\( x_i \)[/tex] is the number of tickets and [tex]\( P(x_i) \)[/tex] is the probability of [tex]\( x_i \)[/tex].
3. Compute each term in the sum:
- For [tex]\( x_1 = 1 \)[/tex]:
[tex]\[
1 \cdot 0.29 = 0.29
\][/tex]
- For [tex]\( x_2 = 2 \)[/tex]:
[tex]\[
2 \cdot 0.44 = 0.88
\][/tex]
- For [tex]\( x_3 = 3 \)[/tex]:
[tex]\[
3 \cdot 0.19 = 0.57
\][/tex]
- For [tex]\( x_4 = 4 \)[/tex]:
[tex]\[
4 \cdot 0.06 = 0.24
\][/tex]
- For [tex]\( x_5 = 5 \)[/tex]:
[tex]\[
5 \cdot 0.02 = 0.10
\][/tex]
4. Sum the results:
[tex]\[
0.29 + 0.88 + 0.57 + 0.24 + 0.10 = 2.08
\][/tex]
Therefore, the expected value of the distribution [tex]\( E(X) \)[/tex] is:
[tex]\[
E(X) = 2.08
\][/tex]
So, the correct answer is 2.1 (rounded to the nearest tenth).
