Question 1173951
Let's break down this problem step-by-step.

**1. Understanding the Problem**

* X is a random variable representing the number of forms (1, 2, 3, 4, 5).
* The probabilities are proportional to X: P(X=x) = cx, where c is a constant.

**a) Finding the Value of the Constant c**

* The sum of all probabilities must equal 1.
* Therefore: P(1) + P(2) + P(3) + P(4) + P(5) = 1
* Substitute P(x) = cx: c(1) + c(2) + c(3) + c(4) + c(5) = 1
* Simplify: c(1 + 2 + 3 + 4 + 5) = 1
* c(15) = 1
* c = 1/15

**b) Probability Distribution**

* Now that we know c = 1/15, we can find the probabilities:
    * P(X=1) = (1/15) * 1 = 1/15
    * P(X=2) = (1/15) * 2 = 2/15
    * P(X=3) = (1/15) * 3 = 3/15 = 1/5
    * P(X=4) = (1/15) * 4 = 4/15
    * P(X=5) = (1/15) * 5 = 5/15 = 1/3

**d) Mean (μ) and Variance (σ^2)**

* **Mean (μ):**
    * μ = E(X) = Σ[x * P(X=x)]
    * μ = (1 * 1/15) + (2 * 2/15) + (3 * 3/15) + (4 * 4/15) + (5 * 5/15)
    * μ = (1 + 4 + 9 + 16 + 25) / 15
    * μ = 55 / 15 = 11/3 ≈ 3.6667

* **Variance (σ^2):**
    * σ^2 = E(X^2) - μ^2
    * First, find E(X^2):
        * E(X^2) = Σ[x^2 * P(X=x)]
        * E(X^2) = (1^2 * 1/15) + (2^2 * 2/15) + (3^2 * 3/15) + (4^2 * 4/15) + (5^2 * 5/15)
        * E(X^2) = (1 + 8 + 27 + 64 + 125) / 15
        * E(X^2) = 225 / 15 = 15
    * Now, calculate σ^2:
        * σ^2 = 15 - (11/3)^2
        * σ^2 = 15 - 121/9
        * σ^2 = (135 - 121) / 9
        * σ^2 = 14 / 9 ≈ 1.5556

**e) Cumulative Distribution Function (CDF)**

* The CDF, F(x) = P(X ≤ x), is the probability that X is less than or equal to x.

* F(1) = P(X ≤ 1) = P(X=1) = 1/15
* F(2) = P(X ≤ 2) = P(X=1) + P(X=2) = 1/15 + 2/15 = 3/15 = 1/5
* F(3) = P(X ≤ 3) = P(X=1) + P(X=2) + P(X=3) = 3/15 + 3/15 = 6/15 = 2/5
* F(4) = P(X ≤ 4) = P(X=1) + P(X=2) + P(X=3) + P(X=4) = 6/15 + 4/15 = 10/15 = 2/3
* F(5) = P(X ≤ 5) = P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5) = 10/15 + 5/15 = 15/15 = 1

* Therefore, the CDF is:

    * F(x) = { 0, if x < 1
               { 1/15, if 1 ≤ x < 2
               { 1/5, if 2 ≤ x < 3
               { 2/5, if 3 ≤ x < 4
               { 2/3, if 4 ≤ x < 5
               { 1, if x ≥ 5