Chapter 9: Continuous Random Variables.

Question 1

What happens to f(x) for a Probability Density Function (Continuous Random Variables)?

Answer 1

You have to do the integral of the functions between the bounds given.

Question 2

What does the integral of a probability density function have to equal for it to be valid?

Answer 2

1.

Question 3

How can the Median of a probability density function be found?

Answer 3

Set an integral between ‘M’ as the upper bound (∫^m) and ‘a’ as the lower bound you are given (∫a) and integrate the function setting it equal to 0.5 –> ∫f(x) = 0.5.

Question 4

How can the Lower Quartile of a probability density function be found?

Answer 4

Set an integral between ‘Q1’ as the upper bound (∫^Q1) and ‘a’ as the lower bound you are given (∫a) and integrate the function setting it equal to 0.25 –> ∫f(x) = 0.25.

Question 5

How can the Upper Quartile of a probability density function be found?

Answer 5

Set an integral between ‘Q3’ as the upper bound (∫^Q3) and ‘a’ as the lower bound you are given (∫a) and integrate the function setting it equal to 0.75 –> ∫f(x) = 0.75.

Question 6

How can the Mode of a probability density function be found?

Answer 6

d[f(x)]/dx. Differentiate the function and set equal to zero and solve.

Question 7

How can the Mean of a probability density function be found?

Answer 7

Set an integral between ‘b’ as the upper bound (∫^b) and ‘a’ as the lower bound you are given (∫a) and integrate the function multiplying f(x) by x –> ∫xf(x).

Question 8

How can the Variance of a probability density function be found?

Answer 8

Set an integral between ‘b’ as the upper bound (∫^b) and ‘a’ as the lower bound you are given (∫a) and integrate the function multiplying f(x) by x² and then taking away the mean² –> ∫x²f(x) - μ².

Question 9

If X and Y are 2 continuous random variables where Y = aX+b, then what is the Expectation - E(Y) - equal to?

Answer 9

E(Y) = E(aX+b) = aE(X) + b.

Question 10

If X and Y are 2 continuous random variables where Y = aX+b, then what is the Variance - Var(Y) - equal to?

Answer 10

Var(Y) = Var(aX+b) = a²Var(X).

Question 11

What formula is the sum of the expectations of 2 independent continuous random variables?

Answer 11

E(X+Y) = E(X) + E(Y).

Question 12

What formula is the sum of the variance’s of 2 independent continuous random variables?

Answer 12

Var(X+Y) = Var(X) + Var(Y).