Stat 130 chapter 6

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Stat 130 – Intro to Math Stat for CS Chapter 6 Reviewer

Jointly Distributed RV’s 1) Joint FMFs for TWO DISCRETE RVs:

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đ?‘ƒ(đ?‘Ľ, đ?‘Ś) = đ?‘ƒ(đ?‘‹ = đ?‘Ľ, đ?‘Œ = đ?‘Ś) đ?‘ƒ[(đ?‘‹, đ?‘Œ)đ??¸đ??´] = ∑(đ?‘‹,đ?‘Œ) ∑đ??¸đ??´ đ?‘ƒ(đ?‘Ľ, đ?‘Ś)

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đ?‘ƒ(đ?‘Ľ, đ?‘Ś) is a joint pmf if it satisfies the ff: a) đ?‘ƒ(đ?‘Ľ, đ?‘Ś) ≼ 0 for all x and y

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Marginal PMFs:

b) ∑đ?‘‹ ∑đ?‘Œ đ?‘ƒ(đ?‘Ľ, đ?‘Ś) = 1 đ?‘ƒđ?‘Ľ (đ?‘Ľ) = ∑đ?‘Œ đ?‘ƒ(đ?‘Ľ, đ?‘Ś)

đ?‘ƒđ?‘Ś (đ?‘Ś) = ∑đ?‘‹ đ?‘ƒ(đ?‘Ľ, đ?‘Ś)

2) Joint PDFs for TWO CONTNUOUS RVs:

đ?‘ƒ[(đ?‘‹, đ?‘Œ)đ??¸đ??´] = âˆŹ

đ?‘“(đ?‘Ľ, đ?‘Ś)đ?‘‘đ?‘Ľđ?‘‘đ?‘Ś đ??´

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If 2d rectangle -> {(đ?‘Ľ, đ?‘Ś): đ?‘Ž ≤ đ?‘Ľ ≤ đ?‘?, đ?‘? ≤ đ?‘Ś ≤ đ?‘‘}, then, đ?‘?

đ?‘‘

đ?‘ƒ[(đ?‘‹, đ?‘Œ)đ??¸đ??´] = đ?‘ƒ(đ?‘Ž ≤ đ?‘‹ ≤ đ?‘?, đ?‘? ≤ đ?‘Œ ≤ đ?‘‘) = âˆŤ âˆŤ đ?‘“(đ?‘Ľ, đ?‘Ś)đ?‘‘đ?‘Śđ?‘‘đ?‘Ľ đ?‘Ž

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đ?‘?

Properties: a) đ?‘“(đ?‘Ľ, đ?‘Ś) ≼ 0, for all x and y âˆ?

�

b) âˆŤâˆ’âˆ? âˆŤâˆ’âˆ? đ?‘“(đ?‘Ľ, đ?‘Ś)đ?‘‘đ?‘Ľđ?‘‘đ?‘Ś = 1 -

Marginal PMFs: �

�

đ?‘“đ?‘Ľ (đ?‘Ľ) = âˆŤâˆ’âˆ? đ?‘“(đ?‘Ľ, đ?‘Ś)đ?‘‘đ?‘Ś

đ?‘“đ?‘Ś (đ?‘Ś) = âˆŤâˆ’âˆ? đ?‘“(đ?‘Ľ, đ?‘Ś)đ?‘‘đ?‘Ľ

Independence of RVs ďƒ˜ Independent iff: a) đ?‘?(đ?‘Ľ, đ?‘Ś) = đ?‘?đ?‘Ľ (đ?‘Ľ)đ?‘?đ?‘Ś (đ?‘Ś) when x and y are discrete b) đ?‘“(đ?‘Ľ, đ?‘Ś) = đ?‘“đ?‘Ľ (đ?‘Ľ)đ?‘“đ?‘Ś (đ?‘Ś) when x and y are continuous Conditional Distributions ďƒ˜ cond. pdf of Y given that X=x is:

đ?‘“đ?‘Ś|đ?‘Ľ (đ?‘Ś|đ?‘Ľ) =

đ?‘“(đ?‘Ľ, đ?‘Ś) đ?‘“đ?‘Ľ (đ?‘Ľ)

Expected Values, Covariance, Correlation â„Ž(đ?‘Ľ, đ?‘Ś) = đ?‘Ľđ?‘Ś 1) đ??¸(đ?‘Ľđ?‘Ś) = ∑đ?‘Ľ ∑đ?‘Ś đ?‘Ľđ?‘Ś đ?‘ƒđ?‘Ľ, đ?‘Ś(đ?‘Ľ, đ?‘Ś) đ??¸(đ?‘‹đ?‘Œ) = âˆŹ đ?‘Ľđ?‘Śđ?‘“(đ?‘Ľ, đ?‘Ś)đ?‘‘đ?‘Ľđ?‘‘đ?‘Ś when independent, đ??¸(đ?‘Ľđ?‘Ś) = đ??¸(đ?‘Ľ)đ??¸(đ?‘Ś) 1

2) Covariance - Strong relationship -> there is dependence between the 2 RVs -> if x is large, y is also large strong + rel. (+cov) -> if x is small, y is also small strong + rel. (+cov) -> if x is large, y is small, & vice versa -> strong – rel. (-cov.) -> if x & y are NOT strongly related, + & - products will ____ to cancel each other out, yielding cov=0

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đ??śđ?‘œđ?‘Ł(đ?‘Ľ, đ?‘Ś) = đ??¸(đ?‘Ľđ?‘Ś) − đ??¸(đ?‘Ľ)đ??¸(đ?‘Ś) -> = 0 if independent -> = đ?‘Ł(đ?‘Ľ) if đ?‘?đ?‘œđ?‘Ł(đ?‘Ľ1 đ?‘Ľ)

3) Correlation Coefficient

đ?‘ƒđ?‘Ľđ?‘Ś =

đ??śđ?‘œđ?‘Ł(đ?‘Ľ,đ?‘Ś) √đ?‘Ł (đ?‘Ľ)đ?‘Ł(đ?‘Ś)

Corr ∈ [−1, 1]

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-> 1->perfect +rel. -> near 0 -> absence of relationship -> -1 -> perfect –rel. đ??śđ?‘œđ?‘&#x;đ?‘&#x;(đ?‘Žđ?‘‹ + đ?‘?, đ?‘?đ?‘Œ + đ?‘‘) = đ??śđ?‘œđ?‘&#x;đ?‘&#x;(đ?‘‹, đ?‘Œ); where a,b,c,d are constants đ?‘ƒ = 1 or −1 iff đ?‘Œ = đ?‘Žđ?‘‹ + đ?‘? for some #s a and b with đ?‘Ž ≠0

Statistics and Their Distributions ďƒ˜ RVs are said to form a (simple) random sample of size n if: a) The đ?‘‹đ?‘– ’s are independent RVs b) ___ đ?‘‹đ?‘– has the same prob. Dist. ďƒ˜ Remarks: a) If SRSWR or from an infinite population, conditions are satisfied đ?‘› b) Approx. satisfied if SRSWOR and n<<N (rule of thumb: ≤ 0.5) đ?‘

Statistic ďƒ˜ RV whose value depends on the observed sample & may vary from sample to sample ďƒ˜ Sampling distribution - probability distribution of a statistic - depend on the size of the pop., sample, and the method of choosing the sample - Standard Error -> standard des. of a statistic Normal Population 1. đ?œŽ is known - đ??¸(đ?‘ĽĚ… ) = đ?œ‡ -> sampling dist. has mean equal to population mean

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đ?‘†. đ??¸. (đ?‘ĽĚ… ) =

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đ?‘Łđ?‘Žđ?‘&#x;(đ?‘ĽĚ… ) =

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đ?‘ĽĚ… âˆź đ?‘ (đ?œ‡,

đ?œŽ

√đ?‘› đ?œŽ2

đ?‘› đ?œŽ2 đ?‘›

= 1 (when standardized)

) 2

2. đ?œŽ is unknown - đ??¸(đ?‘ĽĚ… ) = đ?œ‡ đ?‘

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đ?‘†. đ??¸. (đ?‘ĽĚ… ) =

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đ?‘Łđ?‘Žđ?‘&#x;(đ?‘ĽĚ… ) =

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đ?‘ĽĚ… âˆź đ?‘Ą (đ?œ‡, đ?‘› )

√đ?‘› đ?‘ 2 đ?‘›

>1

đ?‘ 2

- Degrees of freedom = n-1 - T-distribution!!! Non-Normal Population Type of Population FINITE

Sampling Method SRSWR

FINITE

SRSWOR

INFINITE

SRSWR SRSWOR

Mean & Variance

đ?œŽđ?‘Ľ 2 đ?œ‡, đ?‘› đ?œŽđ?‘Ľ 2 đ?‘ − đ?‘› đ?œ‡, ( ) đ?‘› đ?‘ −1 đ?œŽđ?‘Ľ 2 đ?œ‡, đ?‘›

Central Limit Theorem ďƒ˜ If n is sufficiently large, that is đ?‘› > 30, we can assume normality! Remarks

1. 2. 3. 4. 5. 6. 7. 8.

đ??¸(đ?‘Ľ1 Âą đ?‘Ľ2 Âą . . . Âąđ?‘Ľđ?‘˜ = đ??¸(đ?‘Ľ2 )Âą. . . Âąđ??¸(đ?‘Ľđ?‘˜ ) đ??¸(đ?‘Žđ?‘Ľđ?‘– + đ?‘?) = đ?‘Žđ??¸(đ?‘Ľđ?‘– ) + đ?‘? đ??¸(đ?‘Žđ?‘Ľ + đ?‘?đ?‘Œ) = đ?‘Žđ??¸(đ?‘Ľ) + đ?‘?đ??¸(đ?‘Ś) đ??¸(đ?‘Ľ1 đ?‘Ľ2 . . . đ?‘Ľđ?‘˜ ) = đ??¸(đ?‘Ľ1 )đ??¸(đ?‘Ľ2 ). . . đ??¸(đ?‘Ľđ?‘˜ ) -> where đ?‘Ľ1 đ?‘Ľ2 , đ?‘Ľ. .. are independent đ?‘Ł(đ?‘Ľ + đ?‘Ś) = đ?‘Ł(đ?‘Ľ) + đ?‘Ł(đ?‘Ś) + 2đ??śđ?‘œđ?‘Ł(đ?‘‹, đ?‘Œ) -> if independent, Cov = 0 đ?‘Ł(đ?‘Ľ − đ?‘Ś) = đ?‘Ł(đ?‘Ľ) + đ?‘Ł(đ?‘Ś) − 2đ??śđ?‘œđ?‘Ł(đ?‘‹, đ?‘Œ) -> if independent, Cov = 0 đ?‘Ł(đ?‘Žđ?‘Ľ + đ?‘?đ?‘Ś) = đ?‘Ž2 đ?‘Ł(đ?‘Ľ) + đ?‘? 2 [đ?‘Ł(đ?‘Ś)] + 2đ?‘Žđ?‘?đ??śđ?‘œđ?‘Ł(đ?‘‹, đ?‘Œ) -> if independent, Cov = 0 If đ?‘‹1 , đ?‘‹2 , . . . , đ?‘‹đ?‘˜ are uncorrelated RVs, then đ?‘Ł(∑đ?‘˜đ?‘–=1 đ?‘Žđ?‘– đ?‘‹đ?‘– = ∑đ?‘˜đ?‘–=1 đ?‘Žđ?‘– 2 đ?‘Ł(đ?‘‹đ?‘– )

CREDITS: Notes by Camille Salazar Encoded by Gerald Roy CampaĂąano

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