Stat 130 chapter 3

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Stat 130 – Intro to Math Stat for CS Chapter 3 Reviewer

Random Variable - A function from Ί into the set ___ of real #s - A function whose value is a real number - Domain: Ί - Counter domain: ____ - X (random var.) Ί - X (elements of the range)

Cumulative Distribution Function (CDF) -> đ??šđ?‘Ľ (.) - Function defined for any ____ x as đ??šđ?‘Ľ (đ?‘Ľ) = đ?‘ƒ(đ?‘‹ ≤ đ?‘Ľ) - Possibility ___ [0, 1] - Non-decreasing ___ - Every RV will have one & any one CDF - If đ??šđ?‘Ľ (.) is the cdf of a RV X, then for any _____ a and b, đ?‘Ž < đ?‘?: 1) đ?‘ƒ(đ?‘Ž < đ?‘‹ ≤ đ?‘?) = đ??šđ?‘Ľ (đ?‘?) − đ??šđ?‘Ľ (đ?‘Ž) 2) đ?‘ƒ(đ?‘‹ < đ?‘Ž) = ____ − đ??šđ?‘Ľ (đ?‘Ľ) = đ??šđ?‘Ľ (đ?‘Žâˆ’ ) 3) đ?‘ƒ(đ?‘Ž ≤ đ?‘‹ ≤ đ?‘?) = đ??šđ?‘Ľ (đ?‘?) − đ??šđ?‘Ľ (đ?‘Žâˆ’ ) 4) đ?‘ƒ(đ?‘Ž ≤ đ?‘‹ < đ?‘?) = đ??šđ?‘Ľ (đ?‘?− ) − đ??šđ?‘Ľ (đ?‘Žâˆ’ ) 5) đ?‘ƒ(đ?‘Ž < đ?‘‹ < đ?‘?) = đ??šđ?‘Ľ (đ?‘?− ) − đ??šđ?‘Ľ (đ?‘Ž) 6) đ?‘ƒ(đ?‘‹ > đ?‘Ž) = 1 − đ??šđ?‘Ľ (đ?‘Ž) 7) đ?‘ƒ(đ?‘‹ ≼ đ?‘Ž) = 1 − đ??šđ?‘Ľ (đ?‘Žâˆ’ ) 8) đ?‘ƒ(đ?‘‹ = đ?‘Ž) = đ??šđ?‘Ľ (đ?‘Ž) − đ??šđ?‘Ľ (đ?‘Žâˆ’ ) -> if continuous, then = 0 - Any function with domain = ______ counter domain = [0,1] that statistics the ff. is a CDF: 1) đ??šđ?‘Ľ (−âˆ?) = _____ đ??šđ?‘Ľ (đ?‘‹) = 0 and đ??šđ?‘Ľ (âˆ?) = ______ đ??šđ?‘Ľ (đ?‘‹) = 1 2) đ??šđ?‘Ľ (. ) is a monotone, nondecreasing function ( đ??šđ?‘Ľ (đ?‘Ž) ≤ đ??šđ?‘Ľ (đ?‘?)đ?‘“đ?‘œđ?‘&#x; đ?‘Žđ?‘›đ?‘˘ đ?‘Ž < đ?‘?) 3) đ??šđ?‘Ľ (. ) is a continuous from the right ( ______ đ??šđ?‘Ľ (đ?‘Ľ + â„Ž) = đ??šđ?‘Ľ (đ?‘‹) for all x )

Discrete Random Variables ďƒ˜ Sample space contains a finite # of sample points -> Discrete sample space ďƒ˜ RV defined over a discrete sample space

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Probability Mass Function (PMF) ďƒ˜ đ?‘ƒđ?‘Ľ (đ?‘Ľ) = đ?‘ƒ(đ?‘‹ = đ?‘Ľ) ďƒ˜ Mass points: values of the discrete RV for which đ?‘ƒđ?‘Ľ (đ?‘Ľ) ≠0 ďƒ˜ Should satisfy the ff properties: 1) đ?‘ƒđ?‘Ľ (đ?‘Ľ) ≼ 0 2) ∑ all possible x đ?‘ƒđ?‘Ľ (đ?‘Ľ) = 1

Continuous Random Variables ďƒ˜ RV is continuous if the set of all possible values of X consists of an interval on the number line

Probability Density Function (PDF) ďƒ˜ Denoted as f(.) ďƒ˜ Satisfies the ff. properties: 1) đ?‘“(đ?‘Ľ) ≼ 0 for all x 2) The area below the curve, đ?‘“(đ?‘Ľ) & above the x-axis is always equal to 1 3) đ?‘ƒ(đ?‘Ž ≤ đ?‘Ľ ≤ đ?‘?) = area bounded by đ?‘“(đ?‘Ľ), x-axis x=a, and x=b ďƒ˜ Remarks: 1) Graph of PDF is always above the x-axis (cannot take negative value) 2) Area is always equal to 1 with x=a & x=b 3) If x is a continuous RV, đ?‘ƒ(đ?‘Ľ < đ?‘Ž) = đ?‘ƒ(đ?‘Ľ ≤ đ?‘Ž) always! (bec đ?‘ƒ(đ?‘Ľ = đ?‘Ž) = 0) 4) Whenever x is a continuous RV, đ?‘ƒ(đ?‘Ž < đ?‘Ľ < đ?‘?) = đ?‘ƒ(đ?‘Ž ≤ đ?‘Ľ < đ?‘?) = đ?‘ƒ(đ?‘Ž < đ?‘Ľ ≤ đ?‘?) = đ?‘ƒ(đ?‘Ž ≤ đ?‘Ľ ≤ đ?‘?) 5) A RV x is void to have a PDF if

đ?‘‘đ??š(đ?‘Ľ) đ?‘‘đ?‘Ľ

exists for x

đ?‘ƒđ??ˇđ??š =

đ?‘‘(đ??śđ??ˇđ??š) đ?‘‘đ?‘Ľ

6) CDF for a cont. RV X with pdf đ?‘“(đ?‘Ľ) is defined for every y by: đ?‘Ś

đ??š(đ?‘Ś) = đ?‘ƒ(đ?‘Ľ ≤ đ?‘Ś) = âˆŤ đ?‘“(đ?‘Ľ)đ?‘‘đ?‘Ľ −âˆ? *đ??š(đ?‘Ś) is the area under the density curve to the LEFT of x.

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Should satisfy: a) đ?‘“(đ?‘Ľ) ≼ 0 for all x âˆ? đ?‘“(đ?‘Ľ)đ?‘‘đ?‘Ľ −âˆ?

b) Area under the entire density curve âˆŤ

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=1

Expected Value & Variance of a RV - “meanâ€? - đ??¸(đ?‘Ľ) = ∑đ?‘— đ?‘‹đ?‘— đ?‘ƒđ?‘— (đ?‘‹đ?‘— ) -> discrete -

đ??¸(đ?‘Ľ) = âˆŤđ?‘… đ?‘Ľđ?‘“đ?‘Ľ(đ?‘Ľ)đ?‘‘đ?‘Ľ -> continuous

- đ?‘‰(đ?‘Ľ) = đ??¸(đ?‘Ľ 2 ) − [đ??¸(đ?‘Ľ)]2 - Properties: x and y are random variables; a and b are constants 1) đ??¸(đ?‘Žđ?‘Ľ + đ?‘?) = đ?‘Žđ??¸(đ?‘Ľ) + đ?‘? - đ??¸(đ?‘Žđ?‘Ľ) = đ?‘Žđ??¸(đ?‘Ľ) - đ??¸(đ?‘?) = đ?‘? 2) đ??¸(đ?‘Ľ Âą đ?‘Ś) = đ??¸(đ?‘Ľ) Âą đ??¸(đ?‘Ś) 3) đ??¸(đ?‘Ľđ?‘Ś) = đ??¸(đ?‘Ľ)đ??¸(đ?‘Ś); if x & y are independent 4) đ??¸[đ?‘Ľ − đ??¸(đ?‘Ľ)] = 0 5) đ?‘‰(đ?‘Žđ?‘Ľ + đ?‘?) = đ?‘Ž2 đ?‘‰(đ?‘Ľ) - đ?‘‰(đ?‘Žđ?‘Ľ) = đ?‘Ž2 đ?‘‰(đ?‘Ľ) - đ?‘‰(đ?‘?) = 0 6) đ?‘‰(đ?‘Ľ Âą đ?‘Ś) = đ?‘‰(đ?‘Ľ) + đ?‘‰(đ?‘Ś); x & y are independent

CREDITS: Notes by Camille Salazar Encoded by Gerald Roy CampaĂąano

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