程序代写代做 C Probability Distributions

Probability Distributions
Probability Distributions
Discrete
Probability Distributions
Continuous
Probability Distributions
Uniform
Normal
Chi-Sq
Bernoulli Binomial
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F..
Pearson

Bernoulli Distribution
 Consider only two outcomes: “success” or “failure”
 Let P denote the probability of success
 Let 1 – P be the probability of failure
 Define random variable X:
x = 1 if success, x = 0 if failure
 Then the Bernoulli probability distribution is
P(0)  (1P) and P(1)  P
3

Mean and Variance of a Bernoulli Random Variable
 The mean is μx = P
μx E[X]xP(x)(0)(1P)(1)PP X
 The variance is σ2x = P(1 – P)
σ2 E[(Xμ )2](xμ )2P(x)
xx (0P)2(1P)(1P)2PP(1P)
x X
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Ch. 4-4
4
as Prentice Hall

Developing the Binomial Distribution
 The number of sequences with x successes in n independent trials is:
Cnx n! x!(nx)!
Wheren!=n·(n–1)·(n–2)·…·1 and 0!=1
 These sequences are mutually exclusive, since no two can occur at the same time

Binomial Probability Distribution
 A fixed number of observations, n
 e.g., 15 tosses of a coin; ten light bulbs taken from a warehouse
 Two mutually exclusive and collectively exhaustive categories
 e.g., head or tail in each toss of a coin; defective or not defective light bulb
 Generally called “success” and “failure”
 Probability of success is P , probability of failure is 1 – P
 Constant probability for each observation
 e.g., Probability of getting a tail is the same each time we toss the coin
 Observations are independent
 The outcome of one observation does not affect the outcome of
the other
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Pearson

Possible Binomial Distribution Settings
 A manufacturing plant labels items as either defective or acceptable
 A firm bidding for contracts will either get a contract or not
 A marketing research firm receives survey responses of “yes I will buy” or “no I will not”
 New job applicants either accept the offer or reject it
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Ch. 4-7
Pearson

The Binomial Distribution
n!
P(x)  x !(n  x)!P (1- P)
X n X
Pprobability of x successes in n trials, with probability of success P on each
trial
x = number of ‘successes’ in sample, (x = 0, 1, 2, …, n)
n = sample size (number of independent trials or observations)
P = probability of “success”
Example: Flip a coin four times, let x = # heads:
n= 4
P = 0.5
1 – P = (1 – 0.5) = 0.5 x = 0, 1, 2, 3, 4
Ch. 4-8

Shape of Binomial Distribution
 The shape of the binomial distribution depends on the values of P and n
 Here, n = 5 and P = 0.1
 Here, n = 5 and P = 0.5
n=5 P=0.1
0x 012345
P(x) .6
.4 .2
P(x) .6
.4 .2 0
0
n = 5
P = 0.5
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x 12345
Ch. 4-9
as Prentice Hall

Mean and Variance of a Binomial Distribution
 Mean
μE(x)nP
 Variance and Standard Deviation
σ2 nP(1-P) σ  nP(1- P)
Where n = sample size
P = probability of success
(1 – P) = probability of failure
Copyright © 2013
Ch. 4-10
Pearson

Probability Distributions Continuous Random Variables
11

The Normal Distribution
 ‘Bell Shaped’
 Symmetrical
 Mean, Median and Mode are Equal
Location is determined by the mean, μ
Spread is determined by the standard deviation, σ
The random variable has an infinite theoretical range:
+  to  
f(x)
σ
μ
Mean = Median = Mode
x
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The Normal Distribution
 The normal distribution closely approximates the probability distributions of a wide range of random variables
 Distributions of sample means approach a normal distribution given a “large” sample size
 Computations of probabilities are direct and elegant
 The normal probability distribution has led to good business decisions for a number of applications
Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

The Normal Distribution Shape
f(x)
Changing μ shifts the distribution left or right.
Changing σ increases or decreases the spread.
σ
μx
X~N(μ,σ2)
Given the mean μ and variance σ2 we define the
normal distribution using the notation
Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 5-14

The Normal Probability Density Function
 The formula for the normal probability density function is
f(x)  1 e(xμ)2 /2σ2 2π 2
pproximated by 2.71828 pproximated by 3.14159
variable,  < x <  Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall h. 5-15 a a C Cumulative Normal Distribution  For a normal random variable X with mean μ and variance σ2 , i.e., X~N(μ, σ2), the cumulative distribution function is F(x0)P(Xx0) f(x) P(Xx0) μx0 x Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Finding Normal Probabilities The probability for a range of values is measured by the area under the curve P(aXb)F(b)F(a) aμb x Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Finding Normal Probabilities F(b)P(Xb) F(a)P(Xa) aμb x aμb P(aXb)F(b)F(a) Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall x aμb x h. 5-18 C The Standard Normal Distribution  Any normal distribution (with any mean and variance combination) can be transformed into the standardized normal distribution (Z), with mean 0 and f(Z) variance 1 Z~N(0,1) 1 0 Z Z Xμ σ Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Need to transform X units into Z units by Example  If X is distributed normally with mean of 100 and standard deviation of 50, the Z value for X = 200 is Z  X  μ  200  100  2.0 σ 50  This says that X = 200 is two standard deviations (2 increments of 50 units) above the mean of 100. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Comparing X and Z units 100 200 X 0 2.0 Z (μ = 100, σ = 50) (μ=0, σ=1) Note that the distribution is the same, only the scale has changed. We can express the problem in original units (X) or in standardized units (Z) Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall f(x) Finding Normal Probabilities P(a  X  b)  P a  μ  Z  b  μ  σ σ FbμFaμ  σ   σ  aμb aμ 0 bμ σσ x Z h. 5-22 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall C Probability as Area Under the Curve The total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below f(X) P(  X  μ)  0.5 0.5 μ P(μ  X  )  0.5 h. 5-23 P(  X  )  1.0 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 0.5 X C Appendix Table 1  The Standard Normal Distribution table in the textbook (Appendix Table 1) shows values of the cumulative normal distribution function  For a given Z-value a , the table shows F(a) (the area under the curve from negative infinity to a ) Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 0 F(a)P(Za) aZ General Procedure for Finding Probabilities To find P(a < X < b) when X is distributed normally:  Draw the normal curve for the problem in terms of X  Translate X-values to Z-values  Use the Cumulative Normal Table Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Finding Normal Probabilities  Suppose X is normal with mean 8.0 and standard deviation 5.0  Find P(X < 8.6) 8.0 8.6 X Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Finding Normal Probabilities  Suppose X is normal with mean 8.0 and standard deviation 5.0. Find P(X < 8.6) X  μ 8.6  8.0 Z σ  5.0 0.12 (continue d) 8 8.6 P(X < 8.6) X μ=8 μ=0 σ = 10 σ = 1 0 0.12 Z P(Z < 0.12) Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 5-27 Finding the X value for a Known Probability  Steps to find the X value for a known probability: 1. Find the Z value for the known probability 2. Convert to X units using the formula: XμZσ Copyr ight © 2013 Pears on Educ ation, Inc. h. 5-28 Publi C Assessing Normality  Not all continuous random variables are normally distributed  It is important to evaluate how well the data is approximated by a normal distribution Copyr ight © 2013 Pears on Educ ation, Inc. h. 5-29 Publi C The Normal Probability Plot  Normal probability plot ► Arrange data from low to high values ► Find cumulative normal probabilities for all values ► Examine a plot of the observed values vs. cumulative probabilities (with the cumulative normal probability on the vertical axis and the observed data values on the horizontal axis) ► Evaluate the plot for evidence of linearity Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall The Normal Probability Plot Left-Skewed Right-Skewed 100 0 Data 100 0 Data Nonlinear plots indicate a deviation from normality Uniform h. 5-31 100 0 Data Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Percent Percent Percent C h. 5-32 5.6  Let X1, X2, . . ., Xk be continuous random variables  Their joint cumulative distribution function, F(x1, x2, . . ., xk) simultaneously X1 is less than x1, X2 is less than x2, and so on; that is Jointly Distributed Continuous Random Variables F(x,x ,,x )P(X x X x X x ) defines the probability that 12k1122kk Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall C 5.6  Let X1, X2, . . ., Xk be continuous random variables  Their joint cumulative distribution function, F(x1, x2, . . ., xk) simultaneously X1 is less than x1, X2 is less than x2, and so on; that is Jointly Distributed Continuous Random Variables F(x,x ,,x )P(X x X x X x ) defines the probability that 12k1122kk h. 5-33 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall C Sampling Distribution of s2 (Normal Data)  Population variance (2) is a fixed (unknown) parameter based on the population of measurements  Sample variance (s2) varies from sample to sample (just as sample mean does)  Sums of squared indepedet standard Normal Distributions  Chi-Square distributions ► Positively skewed with positive density over (0,) ► Indexed by its degrees of freedom (df) ► Mean=df, Variance=2(df) ► Critical Values given in Table 7, pp. 1095-1096 Chi-Square Distributions 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 Chi-Square Distributions df=4 df=10 df=20 f1(y) f2(y) f3(y) f4(y) f5(y) df=30 df=50 0 10 20 30 40 50 60 70 X^2 f(X^2) Chi-Square Distribution Critical Values 0.12 0.1 0.08 0.06 0.04 .025 0.02 .95 Chi-Square Distribution (df=10) a c 2(a ) df=10 0.995 2.156 0.990 2.558 0.975 3.247 0.950 3.940 0.900 4.865 0.100 15.987 0.050 18.307 0.025 20.483 0.010 23.209 0.005 25.188 0 .025 0 5 10 15 20 25 30 35 40 -0.02 3.247 X^2 20.48 f(X^2) Chi-Square Critical Values (2-Sided Tests/CIs) a/2 1-a a/2 c2L c2U f(X^2) F-Distributions  F distribution is the ratio of (y1/d1)/(y2/d2) where y1 and y2 are Chi-sq distributions with d1 and d2 degrees of freedom  Take on positive density over the range (0 , )  Cannot take on negative values  Non-symmetric (skewed right)  Indexed by two degrees of freedom (d1 (numerator df) and d2 (denominator df)) Properties of F-Distributions F-Distributions 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 f(5,5) f(5,10) f(10,20) Density Function of F 0 1 2 3 4 5 6 7 8 9 10 F 0.7 0.6 0.5 0.4 0.3 0.2 0.1 .05 Critical Values of F (df1=5,df2=5) .90 0 0 1 2 3 4 5 6 7 8 9 10 F .05 F(.95,5,5)=1/F(.05,5 F(.05,5,5)=5.05 Density Function of F