Probability Distributions

3/4

Step 1: F(2) = P(X ≤ 2) means we add all probabilities for x ≤ 2. Step 2: F(2) = f(1) + f(2) = 1/4 + 1/2. Step 3: = 1/4 + 2/4 = 3/4. Step 4: Note: F(3) = f(1)+f(2)+f(3) = 3/4 + 1/4 = 1, which confirms our total sums to 1. The answer is 3/4.

1 - F(x)

Step 1: The total probability for all outcomes = 1. Step 2: P(X ≤ x) + P(X > x) = 1. Step 3: Since P(X ≤ x) = F(x), we get P(X > x) = 1 - F(x). Step 4: This is a standard complement rule applied to CDF. Example: if F(3) = 0.7, then P(X > 3) = 1 - 0.7 = 0.3.

1/4

Step 1: For a valid PDF, integral from -∞ to ∞ of f(x)dx = 1. Step 2: Since f(x) = k only for 2 ≤ x ≤ 6, we integrate k from 2 to 6. Step 3: k × (6 - 2) = 1 → k × 4 = 1. Step 4: k = 1/4. Verification: integral from 2 to 6 of (1/4)dx = (1/4)(4) = 1. ✓