What was in the last lecture? Normal distribution A continuous rv with bell-shaped density curve The pdf is given by f(x) = 1 2πσ e (x µ)2 2σ 2, < x < If X N(µ, σ 2 ), E(X) = µ and V (X) = σ 2 Standard normal rv Z: Z N(0, 1) Cdf of standard normal rv Z: Φ(z) = P(Z z) c Mikyoung Jun (Texas A&M) stat211 lecture 10 February 17, 2011 1 / 18
Probabilities using z-curve P(Z 1.5) = Φ( 1.5) = 0.067 P(Z 2.5) = Φ(2.5) = 0.994 Given the above information, can you get P( 1.5 Z 2.5)? Can you get P( 2.5 Z)? Can you get P( 2.5 Z 1.5)? Can you get P( 1.5 Z 1.5)? For any z, what is the relationship between Φ(z) and Φ( z)? c Mikyoung Jun (Texas A&M) stat211 lecture 10 February 17, 2011 2 / 18
Percentiles of the standard normal distribution 99th percentile of standard normal distribution 99 % 2.00 0.00 2.33 Q. What is 99th percentile of the standard normal distribution? Q. What is the 1st percentile of the standard normal distribution? c Mikyoung Jun (Texas A&M) stat211 lecture 10 February 17, 2011 3 / 18
z α : a notation From now on, z α denote a value for which α of the area under the z-curve lies to the right of z α alpha 0 In terms of percentile, z α is ( Do we know z 0.01? What is z 0.5? )th percentile c Mikyoung Jun (Texas A&M) stat211 lecture 10 February 17, 2011 4 / 18
Relationship between normal and standard normal distributions Now we return to the grade example (from the last lecture) and I ask you: For a randomly selected student, what is the probability that his/her score is between 72 and 90? First, let us think about how to connect the grade distribution (N(60, 12 2 )) with standard normal distribution c Mikyoung Jun (Texas A&M) stat211 lecture 10 February 17, 2011 5 / 18
Relationship between normal and standard normal distributions For X normally distributed with mean µ and variance σ 2, we write X N(µ, σ 2 ) Create a new rv, Y = X µ σ We know Y is also normally distributed What is the mean of Y? What is the variance of Y? For X N(µ, σ 2 ), we have X µ σ N(0, 1), i.e. X µ σ = Z c Mikyoung Jun (Texas A&M) stat211 lecture 10 February 17, 2011 6 / 18
Relationship between normal and standard normal distributions Now for the grade example, we know X N(60, 12 2 ) We want P(72 < X < 90) First, note Z = X?? Then, P(72 < X < 90) = P( 72??) = Φ(?) Φ(?) =? < X?? < 90?? ) = P(? < Z < c Mikyoung Jun (Texas A&M) stat211 lecture 10 February 17, 2011 7 / 18
Exercise For a rv X N(10, 4), what is its 5th percentile? (Step 1) How is X related with the standard normal rv Z? (Step 2) Find the 95th percentile of Z: what is z that satisfies Φ(z) = 0.95? (you may need standard normal table) (Step 3) Then what is the 5th percentile of Z? (Step 4) What is the relationship between the 5th percentile of Z and the 5th percentile of X? c Mikyoung Jun (Texas A&M) stat211 lecture 10 February 17, 2011 8 / 18
Empirical rule about normal distribution (Rule 1) Roughly 68% of the values are within 1 SD of the mean Why is this true? Let X N(µ, σ 2 ) The rule says, P(µ σ < X < µ + σ) = 0.68 Note that P(µ σ < X < µ + σ) = P( 1 < X µ σ < 1) How can we find this probability using Φ(z)? c Mikyoung Jun (Texas A&M) stat211 lecture 10 February 17, 2011 9 / 18
Empirical rule about normal distribution (Rule 2) Roughly 95% of the values are within 2 SD of the mean (Rule 3) Roughly 99.7% of the values are within 3 SD of the mean Verify the above two rules c Mikyoung Jun (Texas A&M) stat211 lecture 10 February 17, 2011 10 / 18
Why are we studying the distributions? So far, we have studied quite a lot of distributions: Bernoulli, Binomial, Poisson, uniform, normal... Why are we doing this? c Mikyoung Jun (Texas A&M) stat211 lecture 10 February 17, 2011 11 / 18
Stat 211 grades Suppose a rv X gives the final scores of students in stat 211-50x in a semester The instructor needs to decide on the letter grades for each students To do that, the instructor needs to know the percentiles of the final scores c Mikyoung Jun (Texas A&M) stat211 lecture 10 February 17, 2011 12 / 18
Stat 211 grades stat 211 50x students final grade pdf 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0 20 40 60 80 100 score c Mikyoung Jun (Texas A&M) stat211 lecture 10 February 17, 2011 13 / 18
Total column ozone data Total column ozone level on May 31 1990 density 0.000 0.002 0.004 0.006 0.008 0.010 250 300 350 400 450 Dobson unit c Mikyoung Jun (Texas A&M) stat211 lecture 10 February 17, 2011 14 / 18
Total column ozone data X: total column ozone level (in Dobson unit) measured on May 31, 1990 by satellite The histogram of X does not look like normal curve, but when we take log-transformation, it looks much closer to normal curve A continuous (and nonnegative) rv X is said to have a lognormal distribution if log(x) has a normal distribution Please read the textbook for more information on this distribution c Mikyoung Jun (Texas A&M) stat211 lecture 10 February 17, 2011 15 / 18
More continuous distributions Gamma distribution: for α, β > 0 A continuous rv X is said to have a gamma distribution if its pdf is { 1 Γ(α)β x α 1 e x/β, x 0 f(x) = α 0, otherwise Gamma distribution is used to model the following situation: How long do you expect to wait to have α many events happening, if the event happens following Poisson distribution with rate 1 β E(X) = αβ and V (X) = αβ 2 For more information, please read the text book (note there is one homework problem related to Gamma distribution) c Mikyoung Jun (Texas A&M) stat211 lecture 10 February 17, 2011 16 / 18
Quick Exercise If bolt thread length is normally distributed, what is the probability that the thread length of a randomly selected bolt is: Within 1.5 SDs of its mean value? Use Φ(1.5) = 0.9332 Farther than 2.5 SDs from its mean value? Use Φ(2.5) = 0.9938 Between 1 and 2 SDs from its mean value? Use the empirical rule c Mikyoung Jun (Texas A&M) stat211 lecture 10 February 17, 2011 17 / 18
Quick Exercise The weight distribution of parcels sent in a certain manner is normal with mean value 12 lb and standard deviation 3.5 lb. The parcel service wishes to establish a weight value c beyond which there will be a surcharge. What value of c is such that 99% of all parcels are at least 1 lb under the surcharge weight? What is µ? σ? c 1 is th percentile of N(µ, σ 2 ) c Mikyoung Jun (Texas A&M) stat211 lecture 10 February 17, 2011 18 / 18