Shade the area corresponding to the probability listed, then find the probability.
$P\left(X<7.5\right)$

• A. ; $P\left(X<7.5\right)=0.15$
• B. ; $P\left(X<7.5\right)=0.25$
• C. ; $P\left(X<7.5\right)=0.5625$
• D. ; $P\left(X<7.5\right)=0.5625$

Answer: (D)

Verified step by step guidance

1. Step 1: Identify the type of probability distribution represented in the graph. The graph shows a triangular probability density function (PDF), which is a continuous distribution. The area under the curve represents probabilities.
2. Step 2: Recognize that the problem asks for the probability P(X < 7.5). This corresponds to the shaded area under the curve from the leftmost point (x = 2.5) to x = 7.5.
3. Step 3: Break the shaded area into geometric shapes for calculation. The shaded region forms a triangle. The base of the triangle spans from x = 2.5 to x = 7.5, and the height corresponds to the value of the PDF at x = 7.5.
4. Step 4: Use the formula for the area of a triangle: Area = (1/2) × base × height. The base is (7.5 - 2.5) = 5, and the height can be determined from the graph as 0.15 (the value of the PDF at x = 7.5).
5. Step 5: Calculate the area using the formula. This area represents the probability P(X < 7.5). Substitute the values into the formula: Area = (1/2) × 5 × 0.15. This will give the probability value.