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#### Continuous Random Variables

###### 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)
###### Examples
1. Given that 1. 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
2. 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
3. Find P(X < m)
• P(X < m) can be interpreted as P(-∞ < X < m) • = 1/24(3(3²) – /3 – 3(2²) + /3) = 13/36
###### 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
###### 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 