X	P(X)
-3	0.45
4	0.1
5	0.2
9	0.25
Start by entering the probability distribution into the calculator.

Press "STAT" then ENTER (or select Edit)

Enter the values for X in the first list and the probabilities in the second list.

Then 2nd MODE to exit the list.

Press "STAT" then > to highlight "CALC"

Select 1-var stats

FOr List: List 1 (2nd 1)

FreqList: L2 (2nd 2)

Then calculate

= Mean =

= Standard Deviation =  (round to 2 decimal places)

To find the mean and standard deviation for this probability distribution using a calculator, follow these steps:

1. **Enter the data into your calculator**:
   - Press `STAT` and then press `ENTER` (or select `Edit`).
   - Enter the X-values (-3, 4, 5, 9) into the first list `L1`.
   - Enter the corresponding probabilities (0.45, 0.1, 0.2, 0.25) into the second list `L2`.
   - Press `2nd` then `MODE` to exit the list entry screen.

2. **Calculate 1-variable statistics**:
   - Press `STAT`, then press the right arrow key to highlight `CALC`.
   - Select `1-Var Stats` (usually option 1).
   - For the `List:`, enter `L1` (press `2nd` and then `1`).
   - For the `FreqList:`, enter `L2` (press `2nd` and then `2`).
   - Press `ENTER` to calculate.

3. **Results**:
   - The calculator will display various statistics, including the **mean** and **standard deviation**.

Let's calculate the mean (\(\mu\)) and standard deviation (\(\sigma\)) manually as well to verify:

### Mean Calculation:

\[
\mu = \sum [X \times P(X)] = (-3 \times 0.45) + (4 \times 0.1) + (5 \times 0.2) + (9 \times 0.25)
\]

\[
\mu = (-1.35) + (0.4) + (1) + (2.25) = 2.3
\]

### Standard Deviation Calculation:

\[
\sigma = \sqrt{\sum [P(X) \times (X - \mu)^2]}
\]

Let's compute this.The calculated values are:

- **Mean (μ)**: 2.3
- **Standard Deviation (σ)**: 5.06 (rounded to two decimal places)

Would you like further details or have any questions?

Here are some related questions to expand your understanding:
1. How would the mean change if one of the probabilities was altered?
2. What does the standard deviation tell us about this data set?
3. How do you interpret a negative X-value in terms of probability?
4. Can you calculate variance from the standard deviation?
5. How would this calculation change if all probabilities were the same?

**Tip**: The standard deviation is a measure of how spread out the values are from the mean. A higher value indicates more variability.

The query submitted by the user:
X    P(X)
-3   0.45
4    0.1
5    0.2
9    0.25
Start by entering the probability distribution into the calculator.

Press 'STAT' then ENTER (or select Edit)

Enter the values for X in the first list and the probabilities in the second list.

Then 2nd MODE to exit the list.

Press 'STAT' then > to highlight 'CALC'

Select 1-var stats

For List: List 1 (2nd 1)

FreqList: L2 (2nd 2)

Then calculate

= Mean =

= Standard Deviation = (round to 2 decimal places)

Learn how to calculate the mean (expected value) and standard deviation for a discrete probability distribution using a step-by-step guide. This problem covers key probability concepts, such as calculating the mean using Σ[X * P(X)] and finding the standard deviation. Suitable for students in Grades 10-12.

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