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[h]SL 4.7 Discrete and continuous random variables and their probability distributions
[q] Discrete Random Variables
[a] D.R.V refers to discrete random variables. This simply means we have a random experiment where the outcomes take only discrete values, and each of these have an assigned probability.
The sum of the probabilities always equals 1: \(\sum_{i=1}^{n}P(X=x_{i})=1\).
The x’s can take numerical or non-numerical values.
[q] Expected Value
[a] This is a very similar to the mean it is what you would expect to be the mean if you repeat the experiment many times. So it will work in a similar manner to calculating the mean from a frequency table [\(\frac{\sum_{}f_{i}x_{i}}{n}\) ], but as we have probabilities instead of frequencies, it essentially negates the need for dividing by n. This leaves us with the formula:
\(μ=E(X)=\sum_{}xP(X=x)\)
To explain: this asks you to find the product of then find the sum of these products. x and its probability.
NOTE 1: E(X) is not a probability, so therefore does not need to be between 0 and 1.
NOTE 2: A ‘fair game’ is an experiment where E(X) = 0 , no gain or loss, on average.
NOTE 3: You could also be given a function to calculate the probabilities, instead of a table.
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