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CIE A Level Maths: Probability & Statistics 2復習筆記2.3.1 Probability Density Function

Category: A-level課程, 教材筆記, 福利干貨 Date: 2022年8月18日下午4:07

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

2-3-1-cie-fig1-pdf-graph

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

VnwVIRDH_2-3-1-cie-fig2-we-solution_a

If the graph is easy to draw, then a sketch of f(x) is helpful
- This can highlight useful features such as the graph (and so probabilities) being?symmetrical
- Some p.d.f. graphs lead to common shapes such as?triangles?or?rectangles?whose areas are easy to find, avoiding the need for integration

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

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) .

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)

(b)Find the?exact?value of the mode of?X

(a)Find the median of?X, giving your answer to three significant figures

(b)Find the?exact?value of the mode of?X

Avoid spending too long sketching the graph of? y = f(x), only do this if the graph is straightforward as finding the median and mode by other means can be just as quick

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