To find the standard deviation of the given probability distribution, we can follow these steps:

Step 1: Find the expected value (mean), $E(X)$

The formula for the expected value $E(X)$ is: $E(X) = \sum x \cdot P(X = x)$ Substitute the given values:

$E(X) = (1 \cdot 0.3) + (2 \cdot 0.1) + (3 \cdot 0.2) + (4 \cdot 0.1) + (5 \cdot 0.3)$ $E(X) = 0.3 + 0.2 + 0.6 + 0.4 + 1.5 = 3.0$

Step 2: Find the variance, $\text{Var}(X)$

The variance is given by: $\text{Var}(X) = \sum P(X = x) \cdot (x - E(X))^2$ Now, calculate $(x - E(X))^2$ for each value of $x$:

• For $x = 1$: $(1 - 3)^2 = 4$
• For $x = 2$: $(2 - 3)^2 = 1$
• For $x = 3$: $(3 - 3)^2 = 0$
• For $x = 4$: $(4 - 3)^2 = 1$
• For $x = 5$: $(5 - 3)^2 = 4$

Now, compute the variance: $\text{Var}(X) = (0.3 \cdot 4) + (0.1 \cdot 1) + (0.2 \cdot 0) + (0.1 \cdot 1) + (0.3 \cdot 4)$ $\text{Var}(X) = 1.2 + 0.1 + 0 + 0.1 + 1.2 = 2.6$

Step 3: Find the standard deviation

The standard deviation is the square root of the variance: $\sigma = \sqrt{\text{Var}(X)} = \sqrt{2.6} \approx 1.6$

Final Answer:

The standard deviation is approximately 1.6.

Would you like further details or have any questions?

Here are 5 related questions:

1. How is the variance calculated in a probability distribution?
2. What does the standard deviation represent in terms of the data?
3. How would the standard deviation change if the probabilities were adjusted?
4. What is the relationship between the expected value and the variance?
5. How does the standard deviation help in comparing two different distributions?

Tip: When calculating standard deviation, remember that it measures how spread out the values are from the mean. A higher standard deviation means more spread, while a lower one indicates that the data points are closer to the mean.