S1: Chapter 8 Discrete Random Variables www.drfrostmaths.com Dr J Frost (jfrosttiffin.kingston.sch.uk) Last modified: 12th January 2016 In Chapter 2, we saw that just like in algebra, we can use a variable to represent some quantity, such
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S1: Chapter 8 Discrete Random Variables www.drfrostmaths.com
Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 12th January 2016<br>
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In Chapter 2, we saw that just like in algebra, we can use a variable to represent some quantity, such as height. i.e. It is just like a variable in statistics, except each outcome has now been assigned a probability. “The probability that…<br>
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The height of a person randomly chosen.
The number of cars that pass in the next hour.
The number of countries in the world. No Yes No Yes No Yes This is a continuous random variable. It does not vary, so is not a variable!<br>
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There are two ways to write the mapping from outcomes to probabilities. Probability Functions The “{“ means we have a ‘piecewise function’. This just simply means we choose the function from a list depending on the input. ? Probability Distribution The table form that you know and love. Advantages of probability function:
Can have a rule/expression based on the outcome. Particularly for continuous random variables (in S2), it would be impossible to list the probability for every outcome. More compact. Advantages of distribution:
Probability for each outcome more explicit. ? ?<br>
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Underlying Sample Space { HHH,
HHT,
HTT,
HTH,
THH,
THT,
TTH,
TTT } Probability Distribution ? ? Probability Function ?<br>
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Edexcel S1 May 2012 ?<br>
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5 7 ?<br>
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? ? ? ?<br>
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? ? If X is the number of heads thrown in 2 throws... ? ? ? ? ? ?<br>
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? ? ? a b c d ? ? ? ? ?<br>
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Shoe Size (x) p(x) Shoe Size (x) F(x) 1 It’s just like how we’d turn a frequency graph into a cumulative frequency graph. ?<br>
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Edexcel S1 May 2013 (R) = 0.4 ? ? Edexcel S1 Jan 2013 F(3) = 1, so (27 + k)/40 = 1, ... ? ?<br>
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Q5-8<br>
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Suppose that we throw a single fair die 60 times, and see the following outcomes: But using the actual probabilities of each outcome (i.e. 1/6 for each), what would we expect the average outcome to be?
3.5 ? ? Throw a lot of times. ?<br>
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Find the expected value of the following distributions (in your head!). ? ? ? Bro Tip: Suppose you treated the probabilities as frequencies then found the mean of the ‘frequency table’. What do you notice? Bro Tip: If the distribution is ‘symmetrical’, i.e. both the outcomes and probabilities are symmetrical about the centre, then the expected value is this central value.<br>
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? ?<br>
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We know how to find it for experimental data. How about for a random variable? Mean of the Squares Minus Square of the Mean – ? ? ? ?<br>
Oh dear god, not again... Recap It’s no different with expected values. What do we expect these to be in terms of the original expected value E[X] and the original variance Var[X]?
The random variable Y has mean 2 and variance 9.
Find:
a) E[3Y+1] = 3E[Y] + 1 = 7
c) Var[3Y+1] = 9Var[Y] = 81
e) E[Y2] = Var[Y] + E[Y]2 = 13
f) E[(Y-1)(Y+1)] = E[Y2 – 1] = E[Y2] – 1 = 12 2 5 ? ? ? ? ? ? ? Bro Exam Tip: This has come up in exams multiple times.<br>
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If X is the throw of a fair die, this obviously is its distribution... We call this a discrete uniform distribution. ? You won’t have exam questions on these, but you’ll revisit them in S2. ? ?<br>
