To find the standard deviation of a probability distribution, we need to follow these steps:

Step 1: Calculate the mean (μ) of the distribution.

The mean of a probability distribution is calculated as: $\mu = \sum x \cdot P(x)$ where $x$ represents the values of the random variable and $P(x)$ represents their corresponding probabilities.

From the table:

$\mu = (0 \cdot 0.1) + (1 \cdot 0.3) + (2 \cdot 0.25) + (3 \cdot 0.35)$ $\mu = 0 + 0.3 + 0.5 + 1.05 = 1.85$

Step 2: Calculate the variance.

The variance of a probability distribution is given by: $\sigma^2 = \sum (x - \mu)^2 \cdot P(x)$ Substitute the value of $\mu$ into the formula:

$\sigma^2 = (0 - 1.85)^2 \cdot 0.1 + (1 - 1.85)^2 \cdot 0.3 + (2 - 1.85)^2 \cdot 0.25 + (3 - 1.85)^2 \cdot 0.35$

Now, calculate each term:

• For $x = 0$: $(0 - 1.85)^2 \cdot 0.1 = (3.4225) \cdot 0.1 = 0.34225$
• For $x = 1$: $(1 - 1.85)^2 \cdot 0.3 = (0.7225) \cdot 0.3 = 0.21675$
• For $x = 2$: $(2 - 1.85)^2 \cdot 0.25 = (0.0225) \cdot 0.25 = 0.005625$
• For $x = 3$: $(3 - 1.85)^2 \cdot 0.35 = (1.3225) \cdot 0.35 = 0.462875$

Now sum these results: $\sigma^2 = 0.34225 + 0.21675 + 0.005625 + 0.462875 = 1.0275$

Step 3: Calculate the standard deviation.

The standard deviation is the square root of the variance: $\sigma = \sqrt{\sigma^2} = \sqrt{1.0275} \approx 1.014$

Final Answer:

The standard deviation of the probability distribution is approximately 1.01.

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Would you like a further breakdown or clarification on any part of the solution? Here are some related questions to explore:

1. How would you calculate the expected value of a probability distribution?
2. What are the differences between variance and standard deviation in probability distributions?
3. How do you calculate the mean for a discrete probability distribution?
4. How do you interpret the standard deviation in terms of the spread of a distribution?
5. Can standard deviation be negative for any distribution?

Tip: Always check that the sum of the probabilities in a probability distribution equals 1; this is a necessary condition for it to be valid!