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 = ?(bh)
  • Trapezoids: A = ?(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?

Identify the?limits?of X for a particular problem

• 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?

• 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?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

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?

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) thenExam Tip