random variables and probability distributions random variables - random outcomes corresponding to...
TRANSCRIPT
Random Variables and Probability Distributions
• Random Variables - Random outcomes corresponding to subjects randomly selected from a population.
• Probability Distributions - A listing of the possible outcomes and their probabilities (discrete r.v.s) or their densities (continuous r.v.s)
• Normal Distribution - Bell-shaped continuous distribution widely used in statistical inference
• Sampling Distributions - Distributions corresponding to sample statistics (such as mean and proportion) computed from random samples
Discrete Probability Distributions
• Discrete RV - Random variable that can take on a finite (or countably infinite) set of discontinuous possible outcomes (Y)
• Discrete Probability Distribution - Listing of outcomes and their corresponding probabilities (y , P(y))
1)(1)(0 yallyPyP
Example - Supreme Court Vacancies
• Supreme Court Vacancies by Year 1837-1975
• Y # Vacancies in Randomly selected year
# Vacancies (y) Frequency (# of Years) Proportion (P(y))0 81 81/139=.58271 43 43/139=.30942 14 14/139=.10073 1 1/139=.0072
>3 0 0/139=.0000Total 139 1.0000
Source: R.J. Morrison (1977), “FDR and the Supreme Court: An Example of the Use of Probability Theory in Political History”,
History and Theory, Vol. 16, pp 137-146
Parameters of a P.D.
• Mean (aka Expected Value) - Long run average outcome
)()( yyPYE Standard Deviation - Measure of the “typical” distance of an outcome from the mean
2222 )()()()( yPyyPyYE
Example - Supreme Court Vacanciesy P(y) yP(y) y2P(y)
0 .5827 .0000 .0000
1 .3094 .3094 .3094
2 .1007 .2014 .4028
3 .0072 .0216 .0648
Total 1.0000 .5324 .7770
7025.4936.)5324(.7770.)(
5324.)(
222
yPy
yyP
Normal Distribution• Bell-shaped, symmetric family of distributions• Classified by 2 parameters: Mean () and standard
deviation (). These represent location and spread• Random variables that are approximately normal have
the following properties wrt individual measurements:– Approximately half (50%) fall above (and below) mean
– Approximately 68% fall within 1 standard deviation of mean
– Approximately 95% fall within 2 standard deviations of mean
– Virtually all fall within 3 standard deviations of mean
• Notation when Y is normally distributed with mean and standard deviation :
),(~ NY
Normal Distribution
95.0)22(68.0)(50.0)( YPYPYP
Example - Heights of U.S. Adults
• Female and Male adult heights are well approximated by normal distributions: YF~N(63.7,2.5) YM~N(69.1,2.6)
INCHESM
76.5
75.5
74.5
73.5
72.5
71.5
70.5
69.5
68.5
67.5
66.5
65.5
64.5
63.5
62.5
61.5
60.5
59.5
Cases weighted by PCTM
20
10
0
Std. Dev = 2.61
Mean = 69.1
N = 99.23
INCHESF
70.5
69.5
68.5
67.5
66.5
65.5
64.5
63.5
62.5
61.5
60.5
59.5
58.5
57.5
56.5
55.5
Cases weighted by PCTF
20
18
16
14
12
10
8
6
4
2
0
Std. Dev = 2.48
Mean = 63.7
N = 99.68
Source: Statistical Abstract of the U.S. (1992)
Standard Normal (Z) Distribution
• Problem: Unlimited number of possible normal distributions (- < < , > 0)
• Solution: Standardize the random variable to have mean 0 and standard deviation 1
)1,0(~),(~ NY
ZNY
• Probabilities of certain ranges of values and specific percentiles of interest can be obtained through the standard normal (Z) distribution
Standard Normal (Z) Distribution
• Standard Normal Distribution Characteristics:– P(Z 0) = P(Y ) = 0.5000
– P(-1 Z 1) = P(-Y +) = 0.6826
– P(-2 Z 2) = P(-2Y +2) = 0.9544
– P(Z za) = P(Z -za) = a (using Z-table)
a 0.500 0.100 0.050 0.025 0.010 0.005za 0.000 1.282 1.645 1.960 2.326 2.576
Finding Probabilities of Specific Ranges
• Step 1 - Identify the normal distribution of interest (e.g. its mean () and standard deviation () )
• Step 2 - Identify the range of values that you wish to determine the probability of observing (YL , YU), where often the upper or lower bounds are or -
• Step 3 - Transform YL and YU into Z-values:
UU
LL
YZ
YZ
• Step 4 - Obtain P(ZL Z ZU) from Z-table
Example - Adult Female Heights
• What is the probability a randomly selected female is 5’10” or taller (70 inches)?
• Step 1 - Y ~ N(63.7 , 2.5)
• Step 2 - YL = 70.0 YU =
• Step 3 -
UL ZZ 52.2
5.2
7.630.70
• Step 4 - P(Y 70) = P(Z 2.52) = .0059 ( 1/170)
z .00 .01 .02 .032.4 .0082 .0080 .0078 .00752.5 .0062 .0060 .0059 .00572.6 .0047 .0045 .0044 .0043
Finding Percentiles of a Distribution
• Step 1 - Identify the normal distribution of interest (e.g. its mean () and standard deviation () )
• Step 2 - Determine the percentile of interest 100p% (e.g. the 90th percentile is the cut-off where only 90% of scores are below and 10% are above)
• Step 3 - Turn the percentile of interest into a tail probability a and corresponding z-value (zp):– If 100p 50 then a = 1-p and zp = za
– If 100p < 50 then a = p and zp = -za
• Step 4 - Transform zp back to original units:
pp zY
Example - Adult Male Heights
• Above what height do the tallest 5% of males lie above?
• Step 1 - Y ~ N(69.1 , 2.6)
• Step 2 - Want to determine 95th percentile (p = .95)
• Step 3 - Since 100p > 50, a = 1-p = 0.05
zp = za = z.05 = 1.645
• Step 4 - Y.95 = 69.1 + (1.645)(2.6) = 73.4
z .03 .04 .05 .061.5 .0630 .0618 .0606 .05941.6 .0516 .0505 .0495 .04851.7 .0418 .0409 .0401 .0392
Statistical Models
• When making statistical inference it is useful to write random variables in terms of model parameters and random errors
YYY )(
• Here is a fixed constant and is a random variable
• In practice will be unknown, and we will use sample data to estimate or make statements regarding its value
Sampling Distributions and the Central Limit Theorem
• Sample statistics based on random samples are also random variables and have sampling distributions that are probability distributions for the statistic (outcomes that would vary across samples)
• When samples are large and measurements independent then many estimators have normal sampling distributions (CLT):– Sample Mean:
– Sample Proportion:
n
NY,~
n
N)1(
,~^
Example - Adult Female Heights
• Random samples of n = 100 females to be selected• For each sample, the sample mean is computed• Sampling distribution:
)25.0,5.63(100
5.2,5.63~ NNY
• Note that approximately 95% of all possible random samples of 100 females will have sample means between 63.0 and 64.0 inches