To solve this problem, we need to follow a few key steps. Let's break it down step by step.
### Step 1: Define the Random Variable and its Probability Distribution
We are given the random variable [tex]\( X \)[/tex] that takes values [tex]\( 6, 7, 8, 9, 10, 11 \)[/tex], with their corresponding probabilities as shown below:
[tex]\[
\begin{array}{c|cccccc}
X & 6 & 7 & 8 & 9 & 10 & 11 \\
\hline
P(X) & 0.05 & 0.1 & 0.3 & 0.1 & 0.15 & 0.3 \\
\end{array}
\][/tex]
### Step 2: Calculate the Mean (Expected Value) [tex]\(\mu\)[/tex]
The mean (expected value) [tex]\(\mu\)[/tex] of a discrete random variable [tex]\( X \)[/tex] with probabilities [tex]\( P(X) \)[/tex] is calculated using the formula:
[tex]\[
\mu = E(X) = \sum_{i} X_i \cdot P(X_i)
\][/tex]
Using the given values:
[tex]\[
\mu = (6 \cdot 0.05) + (7 \cdot 0.1) + (8 \cdot 0.3) + (9 \cdot 0.1) + (10 \cdot 0.15) + (11 \cdot 0.3)
\][/tex]
Carrying out the calculations:
[tex]\[
\mu = 0.3 + 0.7 + 2.4 + 0.9 + 1.5 + 3.3 = 9.1
\][/tex]
Thus, the mean is:
[tex]\[
\mu = 9.1
\][/tex]
### Step 3: Calculate the Variance [tex]\(\sigma^2\)[/tex]
The variance [tex]\(\sigma^2\)[/tex] of a discrete random variable [tex]\( X \)[/tex] with probabilities [tex]\( P(X) \)[/tex] is given by the formula:
[tex]\[
\sigma^2 = \sum_{i} (X_i - \mu)^2 \cdot P(X_i)
\][/tex]
We already have [tex]\(\mu = 9.1\)[/tex]. Now, we substitute and calculate:
[tex]\[
\sigma^2 = (6 - 9.1)^2 \cdot 0.05 + (7 - 9.1)^2 \cdot 0.1 + (8 - 9.1)^2 \cdot 0.3 + (9 - 9.1)^2 \cdot 0.1 + (10 - 9.1)^2 \cdot 0.15 + (11 - 9.1)^2 \cdot 0.3
\][/tex]
Breaking it down step-by-step, we compute each term:
[tex]\[
(6 - 9.1)^2 = (-3.1)^2 = 9.61
\][/tex]
[tex]\[
(7 - 9.1)^2 = (-2.1)^2 = 4.41
\][/tex]
[tex]\[
(8 - 9.1)^2 = (-1.1)^2 = 1.21
\][/tex]
[tex]\[
(9 - 9.1)^2 = (-0.1)^2 = 0.01
\][/tex]
[tex]\[
(10 - 9.1)^2 = (0.9)^2 = 0.81
\][/tex]
[tex]\[
(11 - 9.1)^2 = (1.9)^2 = 3.61
\][/tex]
Now, multiply each squared term by the corresponding probabilities:
[tex]\[
0.05 \cdot 9.61 = 0.4805
\][/tex]
[tex]\[
0.1 \cdot 4.41 = 0.441
\][/tex]
[tex]\[
0.3 \cdot 1.21 = 0.363
\][/tex]
[tex]\[
0.1 \cdot 0.01 = 0.001
\][/tex]
[tex]\[
0.15 \cdot 0.81 = 0.1215
\][/tex]
[tex]\[
0.3 \cdot 3.61 = 1.083
\][/tex]
Summing these products gives the variance:
[tex]\[
\sigma^2 = 0.4805 + 0.441 + 0.363 + 0.001 + 0.1215 + 1.083 = 2.49
\][/tex]
Rounding [tex]\(\sigma^2\)[/tex] to one decimal place, we get:
[tex]\[
\sigma^2 = 2.5
\][/tex]
Thus, the variance is:
[tex]\[
\sigma^2 = 2.5
\][/tex]
