Question 1191863
Here's the solution:

**1. Define the Random Variable:**

Let X be the random variable representing the additional allowance your mother will give you.

**2. Construct the Probability Distribution:**

| Grade Range | Additional Allowance (X) | Probability (P(X)) |
|---|---|---|
| 95 and above | Php 50 | 0.17 |
| 90 - 94 | Php 40 | 0.15 |
| 85 - 89 | Php 30 | 0.28 |
| 80 - 84 | Php 20 | 0.25 |
| 79 and below | -Php 15 | 0.15 |

**3. Calculate the Expected Value (Mean):**

The expected value E(X) is calculated as the sum of each possible value of X multiplied by its probability:

E(X) = (50 * 0.17) + (40 * 0.15) + (30 * 0.28) + (20 * 0.25) + (-15 * 0.15)
E(X) = 8.5 + 6 + 8.4 + 5 - 2.25
E(X) = 25.65

Therefore, the expected additional allowance is Php 25.65.

**4. Calculate the Variance:**

The variance Var(X) is calculated as the expected value of the squared difference between each value of X and the mean:

Var(X) = Σ[(X - E(X))² * P(X)]

Var(X) = (50 - 25.65)² * 0.17 + (40 - 25.65)² * 0.15 + (30 - 25.65)² * 0.28 + (20 - 25.65)² * 0.25 + (-15 - 25.65)² * 0.15
Var(X) = (24.35)² * 0.17 + (14.35)² * 0.15 + (4.35)² * 0.28 + (-5.65)² * 0.25 + (-40.65)² * 0.15
Var(X) = 593.9225 * 0.17 + 205.9225 * 0.15 + 18.9225 * 0.28 + 31.9225 * 0.25 + 1652.4225 * 0.15
Var(X) = 100.97 + 30.89 + 5.30 + 7.98 + 247.86
Var(X) = 392.99

Therefore, the variance of the additional allowance is 392.99.

**5. Calculate the Standard Deviation:**

The standard deviation SD(X) is the square root of the variance:

SD(X) = √Var(X)
SD(X) = √392.99
SD(X) ≈ 19.82

Therefore, the standard deviation of the additional allowance is approximately Php 19.82.

**Summary of Results:**

* **Expected Value (Mean):** Php 25.65
* **Variance:** 392.99
* **Standard Deviation:** Php 19.82