Question 1198348
**1. Define the Random Variable**

* Let X be the random variable representing the possible values of the random variable. 
* Since X has support containing only two numbers, let's assume those numbers are 'a' and 'b'.

**2. Define the Probability Mass Function (PMF)**

* Let P(X = a) = p 
* Let P(X = b) = 1 - p 

**3. Set up Equations Based on Given Information**

* **Expected Value (E[X]):** 
    * E[X] = a * P(X = a) + b * P(X = b) 
    * 5 = a * p + b * (1 - p) 

* **Variance (Var(X)):**
    * Var(X) = E[X²] - (E[X])² 
    * 3 = E[X²] - 5² 
    * E[X²] = 28 

    * E[X²] = a² * P(X = a) + b² * P(X = b) 
    * 28 = a² * p + b² * (1 - p)

**4. Solve the System of Equations**

* We have two equations and two unknowns (a, b, and p). 
* Solve these equations simultaneously to find the values of 'a', 'b', and 'p'.

**Example Solution**

* **Let's assume:** 
    * a = 2 
    * b = 8 

* **Solve for p using E[X] = 5:** 
    * 5 = 2 * p + 8 * (1 - p)
    * 5 = 2p + 8 - 8p
    * 6p = 3
    * p = 1/2 

* **Verify Var(X) = 3:**
    * E[X²] = 2² * (1/2) + 8² * (1/2) = 34
    * Var(X) = E[X²] - (E[X])² = 34 - 5² = 3 

**Therefore, one possible PMF for X is:**

* P(X = 2) = 1/2
* P(X = 8) = 1/2

**Note:** This is just one possible solution. There might be other combinations of 'a', 'b', and 'p' that also satisfy the given conditions for expected value and variance.