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Wk6_Simulation_4ppuniversity of california,berkeley statistic R language slide

'Wk6_Simulation_4ppuniversity of california,berkeley statistic R language slide'
9/28/12 1 Simulation Probability ? Probability allows us to quan5fy statements about the chance of an event taking place. For example ‐ Flip a fair coin 1. What’s the chance it lands heads? 2. Flip it 4 5mes, what propor5on of heads do you expect? 3. Will you get exactly that propor5on? 4. What happens when you fl ip the coin 1000 5mes? 4 Flips ? In 4 fl ips, we can get 0, 1, 2, 3, or 4 Heads and so the propor5on of Heads can be: 0, 0.25, 0.5, 0.75, or 1 ? We expect the propor5on to be 0.5 ? But, a propor5on of 0.25 is quite likely: There are 16 possible ways for 4 tosses to land, e.g. HHHH, HHHT, HHTH, … Each is equally likely, so the chance of any par5cular sequence of Hs and Ts is 1/16 So chance of 0.25 propor5on is 4/16 HTTT, THTT, TTHT, TTTH 4 Flips ? We can think of the propor5on of Heads in 4 fl ips as a sta5s5c because it summarizes data ? No5ce that it is a random quan5ty – it takes on 5 possible values, each with some probability value 0.00 0.25 0.50 0.75 1.00 chance 1/16 4/16 6/16 4/16 1/16 9/28/12 2 1,000 Flips ? When we fl ip the coin 1,000 5mes, we can get a many diff erent possible propor5ons of Heads, i.e. 0, 0.001, 0.002, 0.003, …, 0.998, 0.999, 1.000 ? It’s highly unlikely that we would get 0 for the propor5on – how unlikely? ? What does the distribu5on of the propor5on of heads in 1000 fl ips look like? 1,000 Flips ? With some advanced math tools, we can fi gure this out. ? But we can also get a good idea using a simula5on. ? In our simula5on we will assume that the chance of Heads is 0.5 and fi nd out what the possible values for the propor5on of heads in 1,000 fl ips looks like ? If we were to carry out an experiment with a coin and get a par5cular propor5on, say 0.37, then we could use this simula5on study to help us understand the results of our experiment. Let’s Try It Let’s Generalize 9/28/12 3 Before we consider the role that simulation can play in helping us understand statistics, let’s take a step back and think about the big picture. We can think of probability theory as complimentary to statistical inference. Distribution Observed data Probability Inference A statistic is often just a function of a random sample, for example the sample mean, the 95th quantile, or the sample proportion. Statistics are often used as estimators of quantities of interest about the distribution, called parameters. Statistics are random variables (since they depend on the sample); parameters are not. In simple cases, we can study the sampling distribution of the statistic analytically. For example, we can prove that under mild conditions the distribution of the sample proportion is close to normal for large sample sizes. In more complicated cases, we turn to simulation. The main idea in a simulation study is to replace the mathematical expression for the distribution with a sample from that distribution. In our example: are independent observations from the same distribution. The distribution has center (mean/expected value)
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