To find the standard deviation of a discrete probability distribution, follow these steps:

Step 1: Calculate the expected value (mean), $\mu$.

The expected value $\mu$ is calculated as:

$\mu = \sum (x \cdot P(X=x))$

Substituting the given values:

$\mu = (4 \cdot 0.2) + (5 \cdot 0.3) + (6 \cdot 0.1) + (7 \cdot 0.1) + (8 \cdot 0.3)$ $\mu = 0.8 + 1.5 + 0.6 + 0.7 + 2.4 = 6.0$

So, the expected value $\mu = 6.0$.

Step 2: Calculate the variance, $\sigma^2$.

The variance $\sigma^2$ is calculated using the formula:

$\sigma^2 = \sum \left( (x - \mu)^2 \cdot P(X=x) \right)$

Substituting the values:

$\sigma^2 = ( (4 - 6)^2 \cdot 0.2 ) + ( (5 - 6)^2 \cdot 0.3 ) + ( (6 - 6)^2 \cdot 0.1 ) + ( (7 - 6)^2 \cdot 0.1 ) + ( (8 - 6)^2 \cdot 0.3 )$

$\sigma^2 = ( (-2)^2 \cdot 0.2 ) + ( (-1)^2 \cdot 0.3 ) + ( (0)^2 \cdot 0.1 ) + ( (1)^2 \cdot 0.1 ) + ( (2)^2 \cdot 0.3 )$

$\sigma^2 = (4 \cdot 0.2) + (1 \cdot 0.3) + (0 \cdot 0.1) + (1 \cdot 0.1) + (4 \cdot 0.3)$

$\sigma^2 = 0.8 + 0.3 + 0 + 0.1 + 1.2 = 2.4$

Step 3: Calculate the standard deviation, $\sigma$.

The standard deviation is the square root of the variance:

$\sigma = \sqrt{\sigma^2} = \sqrt{2.4} \approx 1.5$

Final Answer:

The standard deviation is approximately 1.5 (rounded to one decimal place).

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Would you like further explanation on any of the steps, or do you have any other questions?

Here are some related questions you might find interesting:

1. How do you find the variance of a probability distribution for a continuous random variable?
2. What does it mean when a distribution has a low standard deviation?
3. How do you calculate the coefficient of variation from standard deviation?
4. How does the standard deviation relate to the shape of a probability distribution?
5. How can you find the mean and standard deviation of a binomial distribution?

Tip: Always check the sum of probabilities for a discrete distribution. It should equal 1!