Probability And Uniform Distribution

Concept: Probability - Continuous Random Variables and Probability Density Functions (PDF).

Step 1: Define the relationship between probability and PDF. For a continuous random variable with a probability density function , the probability that falls within a specific interval is given by the definite integral of from to : .

Step 2: Set up the definite integral for the given interval. We need to calculate . Because the entire interval lies within the domain where the function is defined as non-zero, we use . The integral setup is: .

Step 3: Find the antiderivative of the function. First, expand the function inside the integral: . Apply the power rule for integration ( ): .

Step 4: Evaluate the integral at the upper and lower limits. Evaluate at the upper limit ( ): . Evaluate at the lower limit ( ): . Make common denominators: .

Step 5: Subtract to find the final probability. Apply the Fundamental Theorem of Calculus by subtracting the lower limit evaluation from the upper limit evaluation: . Find a common denominator ( ): . $ $