Expected value and variance of a random variable
The discrete random variable and the table of its distribution, expected value, variance and standard deviation, linearity of expectation, the effect of an affine change on the variance, sums of independent variables and the fair game.
- 23 cards
- 6 topics
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Each topic is weighted the way the exam weighs it.
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A few cards from the deck
Random variable: what is it?
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A numerical quantity whose value depends on the outcome of a random experiment: the winnings of a game, the number of heads in two tosses.
It attaches a number to each outcome; it is studied through the values it can take and their probabilities.
A fair six-sided die is rolled. What is the expected value of the number shown?
3.53421
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3.5(1 + 2 + 3 + 4 + 5 + 6) / 6 = 21 / 6 = 3.5. The 21 is the sum of the faces, not the mean: each face must be weighted by its probability 1/6.
For two constants a and b, E(aX + b) = ….
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For two constants a and b, E(aX + b) = aE(X) + b.
Linearity of expectation needs no assumption: E(X + Y) = E(X) + E(Y) for any variables X and Y.
Probability distribution: what is it?
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The table giving each value a random variable can take and the probability of getting it.
The probabilities in the table are non-negative and sum to 1: a missing row is found by difference.