Comprehensive analysis of gamma-hydroxybutyrate and gamma ...
The Gamma and Normal DistributionsThe Gamma and Normal Distributions. 3.2, 3.3. The Gamma...
Transcript of The Gamma and Normal DistributionsThe Gamma and Normal Distributions. 3.2, 3.3. The Gamma...
The Gamma and NormalDistributions
3.2, 3.3
The Gamma DistributionConsider a Poisson process with rate ๐๐:Let a random variable, ๐๐, denote the waiting time until the ๐ผ๐ผth occurrence.
๐๐ follows a Gamma Distribution.
2
The Gamma Function, ฮ
3When n is an integer,
This is the definition of the gamma function
Gamma Distribution X~Gamma(๐ผ๐ผ,๐๐)
๐๐ ๐ฅ๐ฅ = 1ฮ(๐ผ๐ผ)๐๐๐ผ๐ผ
๐ฅ๐ฅ๐ผ๐ผโ1๐๐โ๐ฅ๐ฅ/๐๐, 0 โค x < โ
๐ธ๐ธ[๐๐] = ๐ผ๐ผ๐๐
๐๐๐๐๐๐[๐๐] = ๐ผ๐ผ๐๐2
4
Gamma ExampleCustomers arrive in a shop according to a Poisson process with a mean rate of 20 per hour. What is the probability that the shopkeeper will have to wait more than 10 minutes for the arrival of the 4th customer?
๏ฟฝ10
โ1
ฮ 4 34๐ฅ๐ฅ4โ1๐๐โ๐ฅ๐ฅ/3 ๐๐๐ฅ๐ฅ = 0.57
5
Gamma ExampleCustomers arrive in a shop according to a Poisson process with a mean rate of 20 per hour. What is the probability that the shopkeeper will have to wait more than 10 minutes for the arrival of the 4th customer?
๏ฟฝ10
โ1
ฮ 4 34๐ฅ๐ฅ4โ1๐๐โ๐ฅ๐ฅ/3 ๐๐๐ฅ๐ฅ = 0.57
6
Normal Distribution
Normal Distributionโซ Most important distribution in statisticsโซ Fits many natural phenomena such as IQ,
measurement error, height, etc.โซ A symmetric distribution with a central peak, and tails
that taper off.
8
Normal Distribution โ Empirical Rule
9
In a normal distribution, approximately 68/95/99.7% of the data falls within 1/2/3 standard deviations of the mean.
Normal Distribution X~N(๐๐,๐๐2)
๐๐(๐ฅ๐ฅ) = 12๐๐๐๐2
๐๐โ(๐ฅ๐ฅโ๐๐)2
2๐๐2 , -โ < ๐ฅ๐ฅ < โ
๐ธ๐ธ[๐๐] = ๐๐
๐๐๐๐๐๐[๐๐] = ๐๐2
10
Normal DistributionLet X ~ Normal(๐๐,๐๐2)โซ To find the P[a < X < b], one would need to evaluate
the integral:
๏ฟฝ๐๐
๐๐1
2๐๐๐๐2๐๐โ
(๐ฅ๐ฅโ๐๐)22๐๐2 ๐๐๐ฅ๐ฅ.
โซ A closed-form expression for this integral does not exist, so we need to use numerical integration techniques.
11
Notes about the Normal DistributionThe Normal Distribution is symmetric with a central peak:
โซ P[X > c] = P[X < -c]โซ Mean = Median = Modeโซ Half of the area is to the left/right of 0.
Examples: if ๐๐ ~ ๐๐(0,1)โซ ๐๐[๐๐ โค 0.2] = 0.5 + ๐๐[0 โค ๐๐ โค 0.2]โซ ๐๐[๐๐ โค 0.3] = ๐๐[๐๐ โฅ 0.7]
12
ExamplesLet ๐๐ ~ ๐๐(0,1)
a) Find P[Z >2] (0.0228)b) Find P[ -2 < Z < 2] (0.9544)c) Find P[0 < Z < 1.73] (0.4582)
13
14
Linear Transformation TheoremLet X โผ N(ยต, ฯ2). Then Y = ฮฑX + ฮฒ follows also a normal distribution.
๐๐ โผ ๐๐(ฮฑยต+ฮฒ, ฮฑ2ฯ2) Can convert any normal distribution to standard normal by subtracting mean and dividing sd:
โซ Z = ๐๐โ๐๐๐๐
Using this theorem, we can see that ๐๐ ~ ๐๐(0,1)
15(Recall) Let ๐๐ have mean, ๐ธ๐ธ[๐๐], and variance, ๐๐2.Let Y = ๐๐๐๐ + ๐๐. Then, ๐๐ has mean ๐๐๐ธ๐ธ[๐๐] + ๐๐, and variance ๐๐2๐๐2.
Example
Suppose the mass of Thorโs hammers in kg (he has an infinite number) are distributed X โผ N(10, 32). Find the proportion of Thorโs hammers that have mass larger than 13.4 kg. (if we randomly select a hammer, find the probability that its mass > 13.4 kg).
ans. P[X > 13.4] = P[Z > 1.13] = 0.129216
What is z?โซ The value of z gives the number of standard
deviations the particular value of X lies above or below the mean ยต.
17
Examples
Normal Distribution
1 Cream and Fluttr knows that the daily demand for cupcakes is a random variable which follows the normal distribution with mean 43.3 cupcakes and standard deviation 4.6. They would like to make enough so that there is only a 5% chance of demand exceeding the number of cupcakes made. (How many should they make?)
z=1.645 x = 5119
20
21
2 Suppose again that Thorโs hammers are normally distributed with: E[X] = 10, Var[X] = 9.
Find the 25th percentile of X. (How much mass should a hammer have, in order to have more than 25% of all hammers)
Ans. z = -0.675 ๐๐0.25 = 7.975 22
3 Stapletonโs Auto Park of Urbana believes that total sales for next month will follow the normal distribution, with mean, ๐๐, and a standard deviation, ๐๐= $300,000. What is the probability that Stapletonโs sales will fall within $150000 of the mean next month?
Ans. 0.6915 โ 0.3085 = .38323
24