A Brief Introduction Random Variables Random Variable RV A numeric outcome that results from an experiment For each element of an experiments sample space the random variable can take on exactly one value ID: 535449
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Slide1
Probability and Distributions
A Brief IntroductionSlide2
Random Variables
Random Variable (RV): A numeric outcome that results from an experiment
For each element of an experiment’s sample space, the random variable can take on exactly one value
Discrete Random Variable: An RV that can take on only a finite or countably infinite set of outcomes
Continuous Random Variable: An RV that can take on any value along a continuum (but may be reported “discretely”
Random Variables are denoted by upper case letters (
Y
)
Individual outcomes for RV are denoted by lower case letters (
y
)Slide3
Probability Distributions
Probability Distribution: Table, Graph, or Formula that describes values a random variable can take on, and its corresponding probability (discrete RV) or density (continuous RV)
Discrete Probability Distribution: Assigns probabilities (masses) to the individual outcomes
Continuous Probability Distribution: Assigns density at individual points, probability of ranges can be obtained by integrating density function
Discrete Probabilities denoted by: p(
y
) = P(
Y=y
)
Continuous Densities denoted by: f(
y
)
Cumulative Distribution Function: F(
y
) = P(
Y
≤y
)Slide4
Discrete Probability DistributionsSlide5
Continuous Random Variables and Probability Distributions
Random Variable:
Y
Cumulative Distribution Function (CDF):
F
(
y
)=P(
Y
≤y)Probability Density Function (pdf): f(y)=dF(y)/dyRules governing continuous distributions:f(y) ≥ 0 y P(a≤Y≤b) = F(b)-F(a) =P(Y=a) = 0 aSlide6
Expected Values of Continuous RVsSlide7
Means and Variances of Linear Functions of RVsSlide8
Normal (Gaussian) Distribution
Bell-shaped distribution with tendency for individuals to clump around the group median/mean
Used to model many biological phenomena
Many
estimators
have approximate normal sampling distributions (see Central Limit Theorem
)
Notation: Y~N(
m,s
2) where m is mean and s2 is varianceObtaining Probabilities in EXCEL:To obtain: F(y)=P(Y≤y) Use Function: =NORMDIST(y,m,s,1)Virtually all statistics textbooks give the cdf (or upper tail probabilities) for standardized normal random variables: z=(y-m)/s ~ N(0,1)Slide9
Normal Distribution – Density Functions (
pdf
)Slide10
Integer part and first decimal place of z
Second Decimal Place of zSlide11
Chi-Square Distribution
Indexed by “degrees of freedom (
n
)” X~
c
n
2
Z~N(0,1)
Z
2 ~c12Assuming Independence:Obtaining Probabilities in EXCEL:To obtain: 1-F(x)=P(X≥x) Use Function: =CHIDIST(x,n)Virtually all statistics textbooks give upper tail cut-off values for commonly used upper (and sometimes lower) tail probabilitiesSlide12
Chi-Square DistributionsSlide13
Critical Values for Chi-Square Distributions (Mean=
n
, Variance=2
n
)Slide14
Student’s t-Distribution
Indexed by “degrees of freedom (
n
)”
X~t
n
Z~N(0,1), X~
c
n
2Assuming Independence of Z and X:Obtaining Probabilities in EXCEL:To obtain: 1-F(t)=P(T≥t) Use Function: =TDIST(t,n)Virtually all statistics textbooks give upper tail cut-off values for commonly used upper tail probabilitiesSlide15Slide16
Critical Values for Student’s t-Distributions (
Mean=
0
, Variance=
n/(n-2
))
Var
exists for
n
>2Slide17
F-Distribution
Indexed by 2 “degrees of freedom (
n
1
,n
2
)” W~F
n1,n2
X
1 ~cn12, X2 ~cn22Assuming Independence of X1 and X2:Obtaining Probabilities in EXCEL:To obtain: 1-F(w)=P(W≥w) Use Function: =FDIST(w,n1,n2)Virtually all statistics textbooks give upper tail cut-off values for commonly used upper tail probabilitiesSlide18Slide19
Critical Values for F-distributions P(F ≤ Table Value) = 0.95