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A Little Intro to Statistics

A Little Intro to Statistics. What’s the chance of rolling a 6 on a dice? 1/6 What’s the chance of rolling a 3 on a dice? 1/6 Rolling 11 times and not getting a 6? (5/6) 11 ~ 13.4% Rolling 11 times and not getting a 3? (5/6) 11 ~ 13.4%

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A Little Intro to Statistics

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  1. A Little Intro to Statistics • What’s the chance of rolling a 6 on a dice? 1/6What’s the chance of rolling a 3 on a dice? 1/6Rolling 11 times and not getting a 6? (5/6)11 ~ 13.4%Rolling 11 times and not getting a 3? (5/6)11 ~ 13.4% • In a pile of Organic Chemistry Exams, what’s the chance of seeing a score between 75 and 78%?What about between 0 and 3%? • What’s the chance of cancer cells surviving after being exposed to radiation for a period of time? • Period of time is short. • Period of time is long.

  2. Examples of Different Types of Probability Distribution Uniform(i.e. dice role) Exponential(i.e. survivability) Normal(i.e. exam scores) http://www.itl.nist.gov/div898/handbook/eda/section3/eda366.htm

  3. Sampling from a Distribution Take exam scores that are normally distributed • Take a pile of exams • Randomly pick one • Record the score Do this enough times, and you’ll see a normal distribution Can simulate taking samples with any known distribution.

  4. Recombination2 Locus Example Recombination location Locus b Locus a gamete

  5. 2-Locus Recombination Example • Determine amount of time (backwards, in generations) when an event occurs. t1 sampled from exponential distribution w/ mean 2N/(1+R) where R=4Nr, N is effective population size and r per generation per offspring is recombination rate. Locus a past This comes from doing the statistics of taking into account that two events occur: coalescence and recombination. Locus b t1 present gamete 1 gamete 2

  6. 2-Locus Recombination Example 2. Determine what type of event occurred Here, the events possible are coalescence or recombination. 1 . 1 + R R . 1 + R Pcoalesc = 1/(2N) Precomb = 2r where R=4Nr, N is effective population size and r per generation per offspring is recombination rate. Locus a past Let’s consider that a recombinationevent occurs on gamete 2. Locus b t1 present gamete 1 gamete 2

  7. 2-Locus Recombination Example 3. Repeat. . . Determine tn, then determine type of event. t2 sampled from exponential distribution Pcoalesc = 3 choose 2 /(2N) Say next a coalescenceevent occurs betweengamete 1 and locus a ofgamete 2. Precomb = r Locus a past Locus b CAa t2 t1 present gamete 1 gamete 2

  8. 2-Locus Recombination Example Repeat until you find MRCA (most recent common ancestor) for all samples at all loci.  Note that the genealogies for locus a is different than locus b  Locus a CAb past t3 Locus b CAa t2 t1 present gamete 1 gamete 2

  9. Introduction of Mutations CAb x t3 x CAa x x t2 Locus a t1 Locus b gamete 1 gamete 2 present

  10. Infinite site Model • Now recombinations can occur at any location  Can use same procedure as the two loci, just use appropriate probabilities  Slightly different model: 1 kb region, 999 locations of recombination

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