Question

x | p(x)
3 | 0.05
5 | 0.2
9 | 0.3
11 | 0.45
find the mean and standard deviation of this probability distribution.
give your answer to at least 2 decimal places.
mean:
standard deviation:

Explanation:

Step1: Calculate the Mean (Expected Value)

The formula for the mean \(\mu\) of a probability distribution is \(\mu = \sum x \cdot P(x)\).
We multiply each \(x\) by its corresponding \(P(x)\) and sum the results:

• For \(x = 3\), \(3 \cdot 0.05 = 0.15\)
• For \(x = 5\), \(5 \cdot 0.2 = 1.0\)
• For \(x = 9\), \(9 \cdot 0.3 = 2.7\)
• For \(x = 11\), \(11 \cdot 0.45 = 4.95\)

Now sum these products: \(0.15 + 1.0 + 2.7 + 4.95 = 8.8\)

Step2: Calculate the Variance

The formula for variance \(\sigma^2\) is \(\sigma^2 = \sum (x - \mu)^2 \cdot P(x)\).
First, find \((x - \mu)\) for each \(x\):

• For \(x = 3\), \(3 - 8.8 = -5.8\), \((-5.8)^2 = 33.64\), then \(33.64 \cdot 0.05 = 1.682\)
• For \(x = 5\), \(5 - 8.8 = -3.8\), \((-3.8)^2 = 14.44\), then \(14.44 \cdot 0.2 = 2.888\)
• For \(x = 9\), \(9 - 8.8 = 0.2\), \((0.2)^2 = 0.04\), then \(0.04 \cdot 0.3 = 0.012\)
• For \(x = 11\), \(11 - 8.8 = 2.2\), \((2.2)^2 = 4.84\), then \(4.84 \cdot 0.45 = 2.178\)

Now sum these values to get the variance: \(1.682 + 2.888 + 0.012 + 2.178 = 6.76\)

Step3: Calculate the Standard Deviation

The standard deviation \(\sigma\) is the square root of the variance: \(\sigma = \sqrt{\sigma^2} = \sqrt{6.76} = 2.6\)

Answer:

Mean: \(8.80\)
Standard Deviation: \(2.60\)