Given the probability distributions shown to the​ right, complete the following parts.
a. Compute the expected value for each distribution.
b. Compute the standard deviation for each distribution.
c. What is the probability that x will be at least 3 in Distribution A and Distribution​ B?
d. Compare the results of distributions A and B.
Distribution A
Distribution B
 
x Subscript i
​P(Xequalsx Subscript i​)
x Subscript i
​P(Xequalsx Subscript i​)
0
0.06
0
0.40
1
0.12
1
0.25
2
0.17
2
0.17
3
0.25
3
0.12
4
0.40
4
0.06
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Part 1
a. What is the expected value for distribution​ A?
muequals
  
enter your response here 
​(Type an integer or decimal rounded to three decimal places as​ needed.)

Let's start by calculating the expected value (mean) for Distribution A. The formula for the expected value is:

\[
\mu = \sum (x_i \cdot P(X = x_i))
\]

For Distribution A, the values for \(x_i\) and \(P(X = x_i)\) are as follows:

| \(x_i\) | \(P(X = x_i)\) |
|--------|----------------|
| 0      | 0.06           |
| 1      | 0.12           |
| 2      | 0.17           |
| 3      | 0.25           |
| 4      | 0.40           |

Now we calculate the expected value for Distribution A:

\[
\mu_A = (0 \cdot 0.06) + (1 \cdot 0.12) + (2 \cdot 0.17) + (3 \cdot 0.25) + (4 \cdot 0.40)
\]

Let's compute this:

\[
\mu_A = (0) + (0.12) + (0.34) + (0.75) + (1.60) = 2.81
\]

So, the expected value for Distribution A is:

\[
\mu_A = 2.81
\]

---

Now, let's calculate the expected value for Distribution B using the same formula:

| \(x_i\) | \(P(X = x_i)\) |
|--------|----------------|
| 0      | 0.40           |
| 1      | 0.25           |
| 2      | 0.17           |
| 3      | 0.12           |
| 4      | 0.06           |

\[
\mu_B = (0 \cdot 0.40) + (1 \cdot 0.25) + (2 \cdot 0.17) + (3 \cdot 0.12) + (4 \cdot 0.06)
\]

Let's compute this:

\[
\mu_B = (0) + (0.25) + (0.34) + (0.36) + (0.24) = 1.19
\]

So, the expected value for Distribution B is:

\[
\mu_B = 1.19
\]

---

### Answers:
- **a. Expected value for Distribution A**: \( \mu_A = 2.81 \)
- **b. Expected value for Distribution B**: \( \mu_B = 1.19 \)

Do you want me to proceed with the standard deviation, or do you have any questions so far?

Here are five related questions that expand on this information:
1. How is the variance related to the standard deviation in a probability distribution?
2. How can expected values be interpreted in real-life contexts?
3. What happens to the expected value if all probabilities are equal?
4. What is the significance of a higher expected value in a probability distribution?
5. How do you compute the expected value for a continuous distribution?

**Tip**: The expected value is often referred to as the long-run average or mean when considering repeated trials of a probability experiment.

Given the probability distributions shown to the​ right, complete the following parts. a. Compute the expected value for each distribution. b. Compute the standard deviation for each distribution. c. What is the probability that x will be at least 3 in Distribution A and Distribution​ B? d. Compare the results of distributions A and B.

Learn how to compute the expected value and standard deviation for two given probability distributions (A and B). Understand key concepts of probability, including how to determine the probability of x being at least 3. This guide is suitable for students in Grades 10-12.

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