Probability & Statistics 2A Level

6.2 Linear combinations of random variables

1 of 1 objectives have lessons · 0 studied · 0 secure

Lessons

Scaling and shifting one random variableWhat happens to the mean, variance and standard deviation of X when every value is multiplied by a and then has b added, and why a normal variable stays normal. 9 min, not started
Combining two random variablesThe mean and variance of aX + bY, why variances add even for a difference, and why the variance rule needs independence. 9 min, not started
Several observations versus one value multipliedWhy the total of n separate observations has variance nσ², while n times one observation has variance n²σ², and how to handle totals with several parts. 9 min, not started
Probabilities with combined normal variablesUse the fact that combinations of independent normal variables are normal to find probabilities about totals, and to compare two quantities with P(X > Y) = P(X − Y > 0). 10 min, not started
Adding independent Poisson variablesWhy the sum of independent Poisson variables is Poisson with the parameters added, how to combine this with rescaling an interval, and which combinations are not Poisson. 8 min, not started

Syllabus objectives

1 objective

What Cambridge says you should be able to do, in our words. Codes are the topic number plus the objective’s position.

6.2.1

Use, when solving problems, the results that E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X); E(aX + bY) = aE(X) + bE(Y); Var(aX + bY) = a²Var(X) + b²Var(Y) for independent X and Y; if X has a normal distribution then so does aX + b; if X and Y have independent normal distributions then aX + bY has a normal distribution; and if X and Y have independent Poisson distributions then X + Y has a Poisson distribution.

Proofs of these results are not required.

Taught in: Scaling and shifting one random variable, Probabilities with combined normal variables, Adding independent Poisson variables, Several observations versus one value multiplied, Combining two random variables · 21 questions