13.1.6. Mean and Variance using PDF

Try these first

1. 1.

   What is the expected value of a continuous random variable with PDF f(x) = 3x² for x from 0 to 1?

   Hint

   Use the integral definition of expected value.

2. 2.

   Define variance.

   Hint

   Think about variability in data.

3. 3.

   What is the formula for calculating the mean of a continuous random variable?

   • E[X] = ∫ f(x) dx
   • E[X] = ∫ x f(x) dx
   • E[X] = ∫ (x - µ)² f(x) dx

   Hint

   Focus on what you integrate.

4. 4.

   True or False: Variance can be zero for a random variable.

   • True
   • False

   Hint

   Consider what variance measures.

5. 5.

   A continuous random variable's PDF is given by f(x) = 12x(1-x) for 0 ≤ x ≤ 1. Calculate the mean and variance.

   Hint

   Calculate the mean first and then use it to find the variance.

6. 6.

   For a normal distribution with mean 5 and standard deviation 2, determine E[X²] and its relationship to the variance.

   Hint

   Recall the formulas relating mean and variance.