Find the standard deviation for the given probability distribution.xp(X)x.P(X)x2.P(X)00.21  10.08  20.08  30.36  40.27

1. Break Down the Problem

To find the standard deviation of a probability distribution, we will go through the following steps:

1. Calculate the expected value (mean).
2. Calculate the variance.
3. Find the square root of the variance to get the standard deviation.

2. Relevant Concepts

• Mean (μ): $\mu = \sum (x \cdot P(X))$
• Variance (σ²): $\sigma^2 = \sum (x^2 \cdot P(X)) - \mu^2$
• Standard Deviation (σ): $\sigma = \sqrt{\sigma^2}$

3. Analysis and Detail

Let’s gather the values from the probability distribution:

Step 1: Calculate the Expected Value (μ)

$\mu = \sum (x \cdot P(X)) = 0 + 0.08 + 0.16 + 1.08 + 1.08 = 2.4$

Step 2: Calculate the Variance (σ²)

$\sigma^2 = \sum (x^2 \cdot P(X)) - \mu^2$ Calculating $\sum (x^2 \cdot P(X))$: $\sum (x^2 \cdot P(X)) = 0 + 0.08 + 0.32 + 3.24 + 4.32 = 8.96$ Now, substituting to find variance: $\sigma^2 = 8.96 - (2.4)^2 = 8.96 - 5.76 = 3.2$

4. Verify and Summarize

Calculating the standard deviation: $\sigma = \sqrt{\sigma^2} = \sqrt{3.2} \approx 1.79$

Final Answer

The standard deviation for the given probability distribution is approximately $\sigma \approx 1.79$.