#### Continuous Random Variables

__Probability Density Function__

__Probability Density Function__

- It is a function whose area under the graph represents the probability used for continuous random variables
- It is represented by f(x)

**Conditions:**- The total area is always = 1

- It cannot have negative probabilities, therefore the graph cannot dip below the x-axis; f(x) ≥ 0
- The probability that X lies between a and b is the area from a to b
- P(a < X < b) =
- Outside the given interval f(x) = 0
- P(X = b) is always = 0 as there is no area

**NOTE:**- P(X < b) = P(X ≤ b) as no extra area is added
- The mode of a pdf is its maximum point (stationary point)

- The total area is always = 1

__Examples__

__Examples__

- Given that

- Find the value of k
- The total area must equal 1, hence:
- = 75k –
^{125}/_{3}k – 12k +^{8}/_{3}k = 24k = 1 - ∴ k =
^{1}/_{24}

- = 75k –

- The total area must equal 1, hence:
- Find the mode, m
- Mode is the value which has the greatest probability, hence we are looking for the maximum point of the pdf:
^{d}/_{dx}[kx(6 – x)] = 6k – 2kx

- Finding the maximum point (stationary point):
- 6k – 2kx = 0
- 6k = 2kx
- x = 3
- ∴ mode = 3

- Mode is the value which has the greatest probability, hence we are looking for the maximum point of the pdf:
- Find P(X < m)
- P(X < m) can be interpreted as P(-∞ < X < m)

- =
^{1}/_{24}(3(3²) –^{3³}/_{3}– 3(2²) +^{2³}/_{3}) =^{13}/_{36}

- P(X < m) can be interpreted as P(-∞ < X < m)

- Find the value of k

__Mean and Variance__

__Mean and Variance__

- To calculate mean/expectation:

- To calculate variance:
- First calculate E(X) then E(X²) by

- Substitute information and calculate using
- Var(X) = E(X²) – E(X)²

- First calculate E(X) then E(X²) by

__The Median__

__The Median__

**The Cumulative Distribution Function (cdf)**

- It gives the probability that the value is less than b
- P(X < b) or P(X ≤ b)

- It is represented by F(b)
- It is the integral of f(x)

**Median:**is the value of b for which F(b) = 0.5