CIE A Level Maths: Probability & Statistics 2复习笔记2.3.1 Probability Density Function

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

• For a continuous random variable , it is often possible to model probabilities using a function
  • This function is called a probability density function (p.d.f.)
• The graph does not need to start and end on the x-axis

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

How do I solve problems using the PDF?

• Some questions may ask for justification of the use of a given function for a probability density function

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

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

How do I find quartiles (or percentiles) of a continuous random variable?

• In a similar way, to find the median
  • The lower quartile will be the value L such that P(X ≤ L) = 0.25 or
    P(X ≥ L) = 0.75
  • The upper quartile will be the value U such that P(X ≤ U) = 0.75 or
    P(X ≥ U) = 0.25
• Percentiles can be found in the same way
  • The 15^th percentile will be the value k such that P(X ≤ k) = 0.15 or
    P(X ≥ k) = 0.85

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 PDF?

• 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 f(x)
• If the graph is a curve with a (local) maximum point, the mode can be found by differentiating and solving the equation f'(x) = 0
  • If there is more than one solution to f'(x) =  0 , further work may be needed to deduce which answer is the mode
    • Look for valid values of from the definition of the p.d.f.
    • Use the second derivative (f'' (x) ) to deduce the nature of each stationary point
    • You may need to check the values of f(x) at the endpoints too
• 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

(See E(X) & Var (X) (continuous) for the mean)