Probability

Step 1: Use the property of a probability density function (PDF). For any valid PDF, the total area under the curve must be equal to 1. Mathematically, this is expressed as: Substituting the given function:
Step 2: Simplify the integral using symmetry. The function is an even function because . For any even function , the integral from to can be written as twice the integral from 0 to . For , . So the integral becomes:
Step 3: Evaluate the integral. Evaluating at the limits:
Step 4: Solve for 'a'. Substitute the value of the integral back into the equation from Step 2:

Step 1: Recall the fundamental property of a Probability Density Function (PDF). The total area under the curve of a PDF must be equal to 1.
Step 2: Set up the integral for the given piecewise function. The total area can be found by integrating over the three separate intervals. This can also be solved graphically by calculating the area of the trapezoidal shape.

Area 1 (Triangle, 0 to 2): Base = 2. Height at x=2 is . Area = .
Area 2 (Rectangle, 2 to 4): Width = 2. Height = . Area = .
Area 3 (Triangle, 4 to 6): Base = 2. Height at x=4 is . Height at x=6 is . Area = .

Step 3: Sum the areas and solve for k. Total Area = Area 1 + Area 2 + Area 3