Combining independent random variables: normals and Poissons
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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
- Adding up the uncertainty
- The two rules
- A difference of two normals
Read the transcript of the teaching part
Last time
Before we start: last time, in The Poisson distribution, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.
Adding up the uncertainty
A bag of ten apples. The profit from two kinds of sugar. The gap between two journey times. Each is built from several random quantities. Means add and subtract just as you would expect, but spread behaves differently. Variances add, whether you add or subtract the variables, and the coefficients get squared.
The two rules
For independent X and Y, the mean of aX plus bY is a times the mean of X, plus b times the mean of Y. The variance of aX plus bY is a squared Var X, plus b squared Var Y. Note the plus, even for a difference. The squares remove any minus sign. So the variance of X minus Y is Var X plus Var Y.
A difference of two normals
X is normal with mean 44 and variance 16. Y is normal with mean 30 and variance 9. Both independent. X minus Y is also normal. Its mean is 44 minus 30, which is 14. Its variance is 16 plus 9, which is 25. To find the probability that X minus Y is less than 15, standardise: 15 minus 14, over the standard deviation 5, is 0.2. The probability is phi of 0.2, which is 0.579.
How often it comes up
How often Cambridge has asked each part, counted by ExamYard from every past paper in its question bank: Comparing normal variables · Total of independent normals.
The whole chapter: Linear Combinations of Random Variables.
Chapter 6.2: Linear combinations of random variables
- Combining independent random variables: normals and Poissons 8 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.