Probability density functions: constants, mean, variance and median
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The teaching part (1 min) is free to watch. A free account opens the whole lecture: 5 practice questions, 2 real past-paper questions with the answers explained, the mistakes that lose marks, the full-marks tip, a one-card summary and a quick quiz.
What this lecture covers
- Probability as an area
- What a pdf must do
- Finding a constant
Read the transcript of the teaching part
Last time
Before we start: last time, in Combining independent random variables, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.
Probability as an area
How long will you wait for a bus? It could be 3 minutes, or 3.141 minutes. With infinitely many possible values, you cannot give each one a probability. Instead you draw a curve, and the probability of landing in a range is the area under the curve. That curve is the probability density function, and integration is the tool.
What a pdf must do
A probability density function f of x is never negative. The total area under the curve is one. For a pdf defined on a to b, the integral of f from a to b is one. The probability that X lies between c and d is the integral of f from c to d.
Finding a constant
If f of x equals k x for x from 0 to 2, and zero elsewhere, the area is the integral of k x, which is 2 k. Set that equal to one. So k is one half. For a graph that is a straight line, the area may be a triangle or trapezium. Use half base times height; it is quicker.
How often it comes up
How often Cambridge has asked each part, counted by ExamYard from every past paper in its question bank: Mean and variance of a pdf.
The whole chapter: Continuous Random Variables.
Chapter 6.3: Continuous random variables
- Probability density functions: constants, mean, variance and median 5 min
- Probabilities from a pdf, its properties and symmetry 6 min
Also for A Level Maths 9709 Paper 6
Which topics come up most in A Level Maths 9709 Paper 6 past papers, counted from every paper since 2004.