Stat 130 chapter 4

Page 1

Stat 130 – Intro to Math Stat for CS Chapter 4 Reviewer

Parameter - constant that determines the specific form of the density function Parameter Space - set of all possible values that the parameter can take on Discrete Distributions 1. Discrete Uniform Distribution -

-> đ??¸(đ?‘Ľ) =

đ?‘Ľ âˆź đ??ˇ âˆŞ (đ?‘ ) 1

pmf: đ?‘ƒđ?‘Ľ (đ?‘Ľ) = đ??ź{1,2,...,đ?‘ } 2

2

đ?‘ +1 2

-> đ?‘‰đ?‘Žđ?‘&#x;(đ?‘Ľ) =

đ?‘ 2 −1 12

2. Poisson Distribution - Model for the # of occurances of a certain event (usually rare) in a specified space or time interval - Properties: a) Possible to partition a specified time or space interval into many smaller nonoverlapping subinterval b) # of outcomes in one subinterval is independent of another c) The probability that a single outcome will occur in a very short subinterval is proportional to the length of the subinterval d) The probability that more than 1 outcome will occur in such a shirt subinterval is almost đ?œ™ - đ?‘Ľ âˆź đ?‘ƒ0 (â‹‹)

-

pmf: đ?‘ƒđ?‘Ľ (đ?‘Ľ) =

-

đ??¸(đ?‘Ľ) =â‹‹

;

đ?‘’ −⋋⋋đ?‘Ľ đ?‘Ľ!

đ??ź[đ?‘œ,1,2,...} (đ?‘Ľ) ;

â‹‹> 0

đ?‘Łđ?‘Žđ?‘&#x;(đ?‘Ľ) =â‹‹

3. Binomial Distribution - F -> fixed # of n trials, đ?‘› > 1 - I -> independence - T ->two possible outcomes (success or failure) - S -> same probability of a success for each observation

-

đ?‘Ľ âˆź đ??ľđ?‘– (đ?‘›1 đ?‘ƒ)

-

pmf: đ?‘ƒđ?‘Ľ (đ?‘Ľ) = ( ) đ?‘ƒ đ?‘Ľ (1 − đ?‘ƒ)đ?‘›âˆ’đ?‘Ľ đ??ź{0,1,2,...,đ?‘›} đ?‘Ľ ; 0 ≤ đ?‘ƒ ≤ 1;

-

đ??¸(đ?‘Ľ) = đ?‘›đ?‘?;

đ?‘› đ?‘Ľ đ?‘Łđ?‘Žđ?‘&#x;(đ?‘Ľ) = đ?‘›đ?‘?đ?‘ž -> đ?‘ž = 1 − đ?‘?

1

SRSWR

đ?‘› ∈ đ?‘§+


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