Lecture25 - GEM2900: Understanding Uncertainty...

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Unformatted text preview: GEM2900: Understanding Uncertainty & Statistical Thinking DSAP, NUS, Semester 1, 2008/2009 – 1 GEM2900: Understanding Uncertainty & Statistical Thinking Berwin Turlach statba@nus.edu.sg Department of Statistics and Applied Probability National University of Singapore Reminder GEM2900: Understanding Uncertainty & Statistical Thinking DSAP, NUS, Semester 1, 2008/2009 – 196 The fourth CA is due Friday, 14 November, at 5pm. Please read the relevant announcement on IVLE. I will instruct IVLE to send reminder e-mails on Thursday and Friday morning. Please submit your answers if you do not want to receive these e-mails. Calculating with the normal distribution (cont.) GEM2900: Understanding Uncertainty & Statistical Thinking DSAP, NUS, Semester 1, 2008/2009 – 197 squaresolid Consider a normally distributed random variable X with mean μ and variance σ 2 , and the random variable Z = X- μ σ which is standard normal. squaresolid The for c > P( μ- cσ < X < μ + cσ ) = P(- c < Z < c ) squaresolid For example P( μ- σ < X < μ + σ ) = P(- 1 < Z < 1) ≈ . 682 P( μ- 2 σ < X < μ + 2 σ ) = P(- 2 < Z < 2) ≈ . 954 P( μ- 3 σ < X < μ + 3 σ ) = P(- 3 < Z < 3) ≈ . 997 squaresolid This is sometimes called the 68 — 95 — 99.7 rule. squaresolid If data are normally distributed, roughly 68% ( ≈ 2 3 ) of observations should be within one standard deviation (SD) of the mean; and roughly 95% of observations should be within two SDs of the mean. The 68 — 95 — 99.7 rule GEM2900: Understanding Uncertainty & Statistical Thinking DSAP, NUS, Semester 1, 2008/2009 – 198 Normal probability density function μ - 3 σ μ - 2 σ μ - σ μ μ + σ μ + 2 σ μ + 3 σ Mean μ Standard deviation σ Probability of falling into given region 68% 95% 99.7% Why is the normal distribution so important? GEM2900: Understanding Uncertainty & Statistical Thinking DSAP, NUS, Semester 1, 2008/2009 – 199 Motivating example: sums of dice squaresolid The next five slides show the probabilities of the sums of n = 1 , 2 , 3 , 4 and 5 dice. squaresolid What happens to the shape of the probabilities as the number of dice n increases? squaresolid Amazingly, this will happen with (almost) any random variable ; as long as n is large enough the probabilities for the sum (and mean) will start to follow a normal bell-shaped curve. Why is the normal distribution so important? (cont.) GEM2900: Understanding Uncertainty & Statistical Thinking DSAP, NUS, Semester 1, 2008/2009 – 200 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 values Sum of one die 0.00 0.05 0.10 0.15 Why is the normal distribution so important?...
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Lecture25 - GEM2900: Understanding Uncertainty...

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