28.4.3.1. Direct Integration

Try these first

1. 1.

   What is the formula to find the mean value of a function using direct integration?

   Hint

   Remember the integral symbol and what it represents.

2. 2.

   Define the probability density function (PDF).

   Hint

   Think about how probabilities are assigned across a range.

3. 3.

   What does the integral of a function F(x) multiplied by its probability distribution represent?

   • Variance
   • Mean value
   • Total probability

   Hint

   Consider what you are calculating over the distribution.

4. 4.

   True or False: Direct integration is commonly used in engineering applications.

   • True
   • False

   Hint

   Think about complexity of functions encountered in practice.

5. 5.

   Calculate the mean value of a function defined by $F(x) = x^2$ where x follows a uniform distribution between -1 and 1.

   Hint

   Break the integral into parts and remember to apply the bounds of the uniform distribution.

6. 6.

   Find the mean of a function involving two variables, F(x, y) = xy, where x and y are independent random variables uniformly distributed over [0, 1].

   Hint

   Carefully apply the integration in two dimensions by solving the inner integral first.