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b:head_first_statistics:using_the_normal_distribution

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b:head_first_statistics:using_the_normal_distribution [2025/10/26 22:49] – [exercise] hkimscilb:head_first_statistics:using_the_normal_distribution [2025/10/29 02:12] (current) – [All aboard the Love Train] hkimscil
Line 146: Line 146:
 [1] 0.9406201 [1] 0.9406201
 > pnorm(-1.56, lower.tail = FALSE) > pnorm(-1.56, lower.tail = FALSE)
 +[1] 0.9406201
 +> pnorm(-1.56, 0, 1, lower.tail = F)
 [1] 0.9406201 [1] 0.9406201
 > pnorm(64, 71, sqrt(20.25), lower.tail = FALSE) > pnorm(64, 71, sqrt(20.25), lower.tail = FALSE)
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 <WRAP box> <WRAP box>
 <code> <code>
 +  * 1 
 +pnorm(6, 10, sqrt(4), lower.tail = F) 
 +  * 2 
 +pnorm(0.3, 6.3, sqrt(9), lower.tail = F) 
 +  * 3 
 +0.5 = (v - 2)/sqrt(4) 
 +v-2 = 1 
 +v = 3 
 +  * 4 
 +z = (v - mean) / sd  
 +2 = (20 - mean) / sqrt(16) 
 +mean = 12
 </code> </code>
 </WRAP> </WRAP>
Line 210: Line 223:
 \end{eqnarray*} \end{eqnarray*}
  
-<code>> 1-pnorm(-0.44)+<code> 
 +> 1-pnorm(-0.44)
 [1] 0.6700314 [1] 0.6700314
 >  > 
 +> pnorm(69, 71, sqrt(20.25), lower.tail = F)
 +[1] 0.6716394
 +> 
 +> z <- (69 - 71)/ sqrt(20.25)
 +> z
 +[1] -0.4444444
 +> pnorm(z, lower.tail = F)
 +[1] 0.6716394
 +> 
 +
 </code> </code>
  
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 {{:b:head_first_statistics:pasted:20191114-072427.png}} {{:b:head_first_statistics:pasted:20191114-072427.png}}
  
 +기억: 
 +E(ax + b) = a E(x) + b
 +V(ax + b) = a^2 V(x) + 0
 +
 + 
 ===== Independent Observation  ===== ===== Independent Observation  =====
 Rather than transforming the weight of each adult, what we really need to figure out is <fc #ff0000>the probability distribution for the combined weight of four separate adults</fc>. In other words, we need to work out <fc #ff0000>the probability distribution of four independent observations of X</fc>. Rather than transforming the weight of each adult, what we really need to figure out is <fc #ff0000>the probability distribution for the combined weight of four separate adults</fc>. In other words, we need to work out <fc #ff0000>the probability distribution of four independent observations of X</fc>.
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 > pbinom(29,40, 1/4, lower.tail = F) > pbinom(29,40, 1/4, lower.tail = F)
 [1] 4.630881e-11 [1] 4.630881e-11
 +> dbinom(30:40,  40, 1/4)
 + [1] 4.140329e-11 4.451967e-12 4.173719e-13 3.372702e-14 2.314599e-15
 + [6] 1.322628e-16 6.123279e-18 2.206587e-19 5.806808e-21 9.926167e-23
 +[11] 8.271806e-25
 +> 1 - dbinom(0:29, 40, 1/4)
 + [1] 0.9999899 0.9998659 0.9991284 0.9963200 0.9886534 0.9727683
 + [7] 0.9470494 0.9142704 0.8821219 0.8602926 0.8556357 0.8687597
 +[13] 0.8942786 0.9240975 0.9512055 0.9718076 0.9853165 0.9930901
 +[19] 0.9970569 0.9988641 0.9996024 0.9998738 0.9999637 0.9999905
 +[25] 0.9999978 0.9999995 0.9999999 1.0000000 1.0000000 1.0000000
 +> sum(dbinom(30:40,  40, 1/4))
 +[1] 4.630881e-11
 +> 1 - sum(dbinom(0:29, 40, 1/4))
 +[1] 4.630896e-11
 +> 
 +
 </code> </code>
  
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 <WRAP box> <WRAP box>
-이를 R을 이용하여 구하면, +위를 R에서 해보면 
 <code> <code>
-pbinom(5, 12, 1/2)+> dbinom(0, 12, 1/2) + dbinom(1, 12, 1/2) + dbinom(2, 12, 1/2)  
 +>  + dbinom(3, 12, 1/2) + dbinom(4, 12, 1/2) + dbinom(5, 12, 1/2) 
 +[1] 0.387207
 </code> </code>
 +그러나, R에서는 더 간단한 방법으로 
 <code> <code>
 > pbinom(5, 12, 1/2) > pbinom(5, 12, 1/2)
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 </code> </code>
  
-위는 아래와 같음을 이해해야 한다+그리고, 위의 dbinom으로 하나씩 계산한다고 하더라도 아래처럼 하게 된다
 <code> <code>
 > sum(dbinom(c(0:5),12,1/2)) > sum(dbinom(c(0:5),12,1/2))
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 <wrap #continuity_correction_egs /> <wrap #continuity_correction_egs />
 <WRAP box> <WRAP box>
-X < 3  ----  <wrap spoiler> X < 2.5 </wrap> +X < 3   <wrap spoiler> X < 2.5 </wrap> 
-X > 3  ----  <wrap spoiler> X > 3.5 </wrap> +X > 3   <wrap spoiler> X > 3.5 </wrap> 
-X <_ 3  ----  <wrap spoiler> X < 3.5 </wrap> +X <_ 3   <wrap spoiler> X < 3.5 </wrap> 
-X >_ 3  ----  <wrap spoiler> X > 2.5 </wrap> +X >_ 3   <wrap spoiler> X > 2.5 </wrap> 
-3 <_ X < 10   ----  <wrap spoiler> 2.5 < X < 9.5 </wrap> +3 <_ X < 10    <wrap spoiler> 2.5 < X < 9.5 </wrap> 
-X = 0  ----  <wrap spoiler> -0.5 < X < 0.5 </wrap> +X = 0   <wrap spoiler> -0.5 < X < 0.5 </wrap> 
-3 <_ X <_ 10  ----  <wrap spoiler> 2.5 < X < 10.5 </wrap> +3 <_ X <_ 10   <wrap spoiler> 2.5 < X < 10.5 </wrap> 
-3 < X <_ 10  ----  <wrap spoiler> 3.5 < X < 10.5 </wrap> +3 < X <_ 10   <wrap spoiler> 3.5 < X < 10.5 </wrap> 
-X > 0  ----  <wrap spoiler> X > 0.5 </wrap> +X > 0   <wrap spoiler> X > 0.5 </wrap> 
-3 < X < 10  ----  <wrap spoiler> 3.5 < X < 9.5 </wrap>+3 < X < 10   <wrap spoiler> 3.5 < X < 9.5 </wrap>
 </WRAP> </WRAP>
  
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 {{:b:head_first_statistics:pasted:20191118-113020.png}} {{:b:head_first_statistics:pasted:20191118-113020.png}}
  
-$\lambda > 15$ 일 때, Poisson distribution, $X \sim Po(\lambda)$는 $X \sim N(\lambda, \lambda)$ 의 성격을 취한다.+<fc #ff0000>$\lambda > 15$ 일 때,</fc> Poisson distribution, $X \sim Po(\lambda)$는 $X \sim N(\lambda, \lambda)$ 의 성격을 취한다.
  
 예)  예) 
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 $0.9654916 \sim 0.9656205$ $0.9654916 \sim 0.9656205$
 +
 +R에서 ppois을 이용하면
 +<code>
 +> ppois(51, 40)
 +[1] 0.9612598
 +> 
 +
 +</code>
  
 ===== Check up ===== ===== Check up =====
b/head_first_statistics/using_the_normal_distribution.1761518952.txt.gz · Last modified: by hkimscil
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