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# Lecture_3 - Introduction to Mathematical Statistics Week 2...

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Week 2 Introduction to Mathematical Statistics

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Recap
Axiom 3

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Axiom 3
Recap What’s the pr of getting 2 heads on 3 tosses of a coin? 1) What is done and what is observed? 2) Write out the sample space 3) Define a reasonable probability function 4)Express the event of interest, say A, as a collection of sample points 5)Compute P(A)

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What is the pr of getting 2 heads on 20 tosses of a coin?
Seem tedious?

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Combinatorics – counting tools
2 dice rolls

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Factorials For n a positive integer, we define n! = n(n – 1)(n – 2). ..(3)(2)(1). We also define 0! = 1. Eg 2: 4! = 4(3)(2)(1) = 24.
Result The number of different possible arrangements of n distinct objects in a row is n !. Eg 3. How many different 3-letter words can be formed from the letters A, C, T?

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Permutations
Combinations

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Combinations Another question: How many different committees of size 3 can be selected from 5 people? Eg. 6 A committee of 5 is to be selected randomly from 12 people. What is the probability that it will contain the two oldest people? Eg. 7 Two separate committees of size 3 and 4 are to be selected from 15 people. How many different pairs of committees are possible?
Combinations

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Combinations Eg. 8 What’s the pr of getting 2 H’s on 5 tosses of a coin?
Pascal’s identity leads to Pascal’s triangle : C (0,0) 1 C (1,0) C (1,1) 1 1 C (2,0) C (2,1) C (2,2) C (3,0) 1 C (3,1) 2 C (3,2) 1 C (3,3) 1 3 3 1 C (4,0) C (4,1) C (4,2) C (4,3) C (4,4) 1 4 6 4 1

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Lecture_3 - Introduction to Mathematical Statistics Week 2...

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