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.