4.7.1 Probability Density Function

A continuous random variable can take any value in an interval so is typically used when continuous quantities are involved (time, distance, weight, etc)

What is a probability density function (p.d.f.)?

• For a continuous random variable, a function can be used to model probabilities
• This function is called a probability density function (p.d.f.), denoted by f(x)
• For f(x) to represent a p.d.f. the following conditions must apply
• f(x) ≥ 0 for all values of x
• The area under the graph of y = f(x) must total 1
• In most problems, the domain of x is restricted to an interval, a ≤ X ≤ b say, with all values of x outside of the interval having f(x)=0

How do I find probabilities using a probability density function (p.d.f.)?

• The probability that the continuous random variable X lies in the interval a ≤ X ≤ b, where X has the probability density function f(x), is given by

• • P(a ≤ X ≤ b) = P(a < X < b)
    • For any continuous random variable (including the normal distribution) P(X = n) = 0
    • One way to think of this is that a = b in the integral above
• For linear functions it can be easier to find the probability using the area of geometric shapes
  • Rectangles: A = bh
  • Triangles: A = ½(bh)
  • Trapezoids: A = ½(a+b)h

How do I determine whether a function is a pdf?

• Some questions may ask for justification of the use of a given function for a probability density function
  • In such cases check that the function meets the two conditions
    • f(x) ≥ 0 for all values of x
    • total area under the graph is 1

How do I use a pdf to find probabilities?

e.g.  

• Trickier problems may involve finding a limit of the integral given its value
  • i.e. Find one of the boundaries in the domain of X, given the probability
    • e.g. Find the value of a given that P(0 ≤ X ≤ a) = 0.09

What is meant by the median of a continuous random variable?

• The median, m, of a continuous random variable, X, with probability density function f(x) is defined as the value of X such that

• Since P(X = m) = 0 this can also be written as 
• IF the p.d.f. is symmetrical (i.e. the graph of y = f(x) is symmetrical) then the median will be halfway between the lower and upper limits of x
  • In such cases the graph of y = f(x) has axis of symmetry in the line x = m

How do I find the median of a continuous random variable?

• The median, m, of a continuous random variable, X, with probability density function f(x) is defined as the value of X such that

• The equation that should be used will depend on the information in the question
  • If the graph of y = f(x) is symmetrical, symmetry may be used to deduce the median
  • This may often be the case if f(x) is linear and the area under the graph is a basic shape such as a rectangle

How do I find the median of a continuous random variable with a piecewise p.d.f.?

• For piecewise functions, the location of the median will determine which equation to use in order to find it
  • For example
    • if 
    • then  so the median must lie in the interval 2 ≤ x ≤ 5
    • so to find the median, m, solve 
      ('0.4 of the area' already used for 0 ≤ x ≤ 2)
    • Use a GDC to plot the function and evalutae integral(s)

What is meant by the mode of a continuous random variable?

• The mode of a continuous random variable, X, with probability density function f(x) is the value of x that produces the greatest value of f(x)

How do I find the mode of a continuous random variable?

• This will depend on the type of function f(x); the easiest way to find the mode is by considering the shape of the graph of y = f(x)
• If the graph is a curve with a maximum point, the mode can be found by differentiating and solving f’(x) = 0
  • If there is more than one solution to f’(x) = 0 then further work may be needed in deducing the mode
    • There could be more than one mode
    • Look for valid values of x from the domain of the p.d.f.
    • Use the second derivative (f’’(x)) to deduce the nature of each stationary point
    • Check the values of f(x) at the lower and upper limits of x, one of these could be the maximum value f(x) reaches
• If the graph of y = f(x) is symmetrical, symmetry may be used to deduce the mode
  • For a symmetrical p.d.f., median = mode = mean

What are the mean and variance of a continuous random variable?

• E(X) is the expected value, or mean, of the continuous random variable X
• E(X) can also be denoted by μ
• Var(X) is the variance of the continuous random variable X
• Var(X) can also be denoted by σ²
• The standard deviation, σ, is the square root of the variance

How do I find the mean and variance of a continuous random variable?

• The mean is given by

• • This is given in the formula booklet
  • If the graph of y = f(x) has axis of symmetry, x = a, then E(X) = a
• The variance is given by

where 

• • This is given in the formula booklet
  • Another version of the variance is given in the formula booklet

• • but the first version above is usually more practical for solving problems
• Be careful about confusing E(X²) and [E(X)]²
  •    ^"mean of the squares"
  •               "mean of the squares"

How do I find the mean and variance of a linear transformation of a continuous random variable?

• For the continuous random variable, X, with mean E(X) and variance Var(X) then