To compute the mean and standard deviation of a discrete probability distribution, we can use the following formulas:
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Mean (Expected Value): \[ \mu = E(X) = \sum (x \cdot P(x)) \]
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Standard Deviation: First, we need to calculate the variance using the formula: \[ \sigma^2 = Var(X) = \sum ((x - \mu)^2 \cdot P(x)) \] Then, the standard deviation is the square root of the variance: \[ \sigma = \sqrt{\sigma^2} \]
Step 1: Calculate the Mean
Using the given values for \(x\) and \(P(x)\):
\[ \begin{align*} \mu & = (-4 \cdot 0.24) + (-1 \cdot 0.14) + (6 \cdot 0.27) + (8 \cdot 0.26) + (9 \cdot 0.09) \ & = (-0.96) + (-0.14) + (1.62) + (2.08) + (0.81) \ & = -0.96 - 0.14 + 1.62 + 2.08 + 0.81 \ & = 3.41 \end{align*} \]
Mean:
\[ \mu = 3.41 \]
Step 2: Calculate the Variance
Now we will need to calculate the variance using the mean we found (3.41):
\[ \begin{align*} \sigma^2 & = ((-4 - 3.41)^2 \cdot 0.24) + ((-1 - 3.41)^2 \cdot 0.14) + ((6 - 3.41)^2 \cdot 0.27) + ((8 - 3.41)^2 \cdot 0.26) + ((9 - 3.41)^2 \cdot 0.09) \ & = ((-7.41)^2 \cdot 0.24) + ((-4.41)^2 \cdot 0.14) + ((2.59)^2 \cdot 0.27) + ((4.59)^2 \cdot 0.26) + ((5.59)^2 \cdot 0.09) \ & = (54.9281 \cdot 0.24) + (19.4481 \cdot 0.14) + (6.7081 \cdot 0.27) + (20.3681 \cdot 0.26) + (31.2481 \cdot 0.09) \ & = 13.1887 + 2.7327 + 1.8092 + 5.2957 + 2.8111 \ & = 25.0374 \end{align*} \]
Step 3: Calculate the Standard Deviation
Now we can compute the standard deviation:
\[ \sigma = \sqrt{\sigma^2} = \sqrt{25.0374} \approx 5.0037 \]
Final Answers:
- Mean: \(\mu = 3.410\)
- Standard Deviation: \(\sigma \approx 5.004\)
Thus, after rounding to three decimal places, the answers are:
- Mean: 3.410
- Standard Deviation: 5.004
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