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IB DP Maths: AA HL復(fù)習(xí)筆記4.4.2 Mean & Variance

Category: IB課程, 教材筆記, 福利干貨 Date: 2022年7月13日上午11:13

E(X)?means the?expected value?or the?mean?of a?random variable X
- The expected value does not need to be an obtainable value of?X
- For example: the expected value number of times a coin will land on tails when flipped 5 times is 2.5
For a?discrete?random variable, it is calculated by:
- How can I decide if a game is fair?

Let?X?be the random variable that represents the?gain/loss?of a player in a game
- X?will be?negative?if there is a?loss
Normally the expected gain or loss is calculated by?subtracting?the?cost to play?the game from the?expected value?of the?prize
If E(X) is?positive?then it means the player can?expect to make a gain
If E(X) is?negative?then it means the player can?expect to make a loss
The game is called?fair?if the?expected gain is 0
- E(X) = 0

a)? ? ?Calculate the expected value of Daphne's prize.

4-4-2-ib-ai-aa-sl-expected-values-a-we-solution

b)? ? ?Determine whether the game is fair.

4-4-2-ib-ai-aa-sl-expected-values-b-we-solution

Var(X)?means the variance of a?random variable X
- The?standard deviation?is the?square root?of the variance
  - This provides a?measure of the spread?of the outcomes of?X
- The variance and standard deviation can?never be negative
The variance of X is the?mean of the squared difference?between X and the mean

- This is given in the formula booklet
  - Compare this formula to the formula for the?variance of a set of data
- This formula works for both?discrete?and?continuous?X

E(X2)?means the?expected value?or the?mean?of the?random variable?defined as?X2
For a?discrete?random variable, it is calculated by:
- Squaring?each value of X?to get the values of X²
- Multiplying each value?of X²?which its corresponding?probability
- Adding?all these terms together
  - E(f(X))?can be found in a similar way

Definitely not!
- They are only equal if X can only take one value
E(X2) is the?mean of the values of X2
E(X)2 is the?square of the mean of the values of X
To see the difference
- Imagine a random variable X that can only take 1 and -1 with equal chance
- E(X) = 0 so?E(X)2 = 0
- The square values are 1 and 1 so?E(X2) = 1

In an exam you can enter the probability distribution into your GDC using the statistics mode
- Enter the possible values as the data
- Enter the probabilities as the frequencies
You can then calculate the mean and variance just like you would with data

4-4-2-ib-aa-hl-variance-we-solution

Suppose?X?is?transformed?by the function?f?to form a new variable?T?=?f(X)
- This means the function?f?is applied to all possible values of?X

If?a?and?b?are constants then the following formulae are true:
- E(aX?±?b) =?aE(X) ± b
- Var(aX ±b) =?a2 Var(X)
  - These are given in the?formula booklet
This is the same as linear transformations of data
- The mean is affected by multiplication and addition/subtraction
- The variance is affected by multiplication but not addition/subtraction
Remember division can be written as a multiplication