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--- r:probability [2016/10/19 01:29] – hkimscil
+++ r:probability [2026/04/14 23:24] (current) – [Normal distribution functions] hkimscil
@@ Line 1: / Line 1: @@
+====== Normal distribution functions ======
 ^ Function  ^ Purpose  ^
@@ Line 21: / Line 22: @@
 | Chi-squared (Chisquare)  | chisq  | df = degrees of freedom  |
 | Exponential  | exp  | rate  |
-| F  | f  | df1 and df2 = degrees of freedom  |
+| <fc #ff0000>**F**</fc>  | f  | df1 and df2 = degrees of freedom  |
 | Gamma  | gamma  | rate; either rate or scale  |
 | Log-normal (Lognormal)  | lnorm  | meanlog = mean on logarithmic scale; \\ sdlog = standard deviation on logarithmic scale   |
 | Logistic  | logis  | location; scale  |
 | Normal  | norm  | mean; \\ sd = standard deviation  |
-| Student’s t (TDist)  | t  | df = degrees of freedom  |
+| <fc #ff0000>**Student’s t (TDist)**</fc>  | t  | df = degrees of freedom  |
 | Uniform  | unif  | min = lower limit; \\ max = upper limit  |
 | Weibull  | weibull  | shape; scale  |
 | Wilcoxon  | wilcox  | m = number of observations in first sample; \\ n = number of observations in second sample   |
+===== pnorm, qnorm =====
+<WRAP info>
+Normal distribution
+$ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{\frac{-(x-\mu)^2}{2\sigma^2}} $
+Assume that the test scores of a college entrance exam fits a normal distribution. Furthermore, the mean test score is 72, and the standard deviation is 15.2. What is the percentage of students scoring 84 or more in the exam?
+<code>> pnorm(72, mean=72, sd=15.2, lower.tail=FALSE)
+[1] 0.5
+> pnorm(1.96)
+[1] 0.9750021
+> pnorm(1.96)-pnorm(-1.96)
+[1] 0.9500042
+> pnorm(c(1.96, -1.96))
+[1] 0.9750021 0.0249979
+> pnorm(84, mean=72, sd=15.2, lower.tail=FALSE)
+[1] .2149176
+> qnorm(.2149176, mean=72, sd=15.2, lower.tail=FALSE)
+[1] 84
+</code></WRAP>
+===== rnorm =====
+Random samples from a normal distribution
+<code>> set.seed(1024)
+> rnorm(50)
+ [1] -0.778662882 -0.389476396 -2.033798329 -0.982373104  0.247890054
+ [6] -2.103864629 -0.381418049  2.074919838  1.027138407  0.473014228
+[11] -1.879263193 -1.239189026  1.160418602  0.003671291 -0.095452066
+[16]  1.795551228 -1.322138481 -0.276086413 -0.743976510 -1.070050125
+[21] -0.349525474  0.805559661  1.605301660  1.447595754 -0.128302224
+[26] -0.538926447  0.391586050  0.879217023 -0.824732092  0.732876423
+[31] -0.664914510  0.360885549  1.011930957 -0.235916848  1.353589893
+[36] -0.268632965  1.019877368 -0.279706500 -0.618146278 -0.499273059
+[41] -0.153716777  1.220869694 -0.669570510 -1.209660342  1.024096655
+[46]  0.603955311 -0.568653469 -0.891303117 -2.525145692  0.589357049</code>
+===== qt, pt =====
+<WRAP info>
+$t = \frac{Z}{\sqrt{\frac{V}{m}}}$
+<code>> qt(c(0.025, 0.975), df=5)
+[1] -2.5706  2.5706
+> qt(c(0.025, 0.975), df=10)
+[1] -2.228139  2.228139
+> qt(c(0.025, 0.975), df=20)
+[1] -2.085963  2.085963
+> qt(c(0.025, 0.975), df=30)
+[1] -2.042272  2.042272
+> qt(c(0.025, 0.975), df=40)
+[1] -2.021075  2.021075
+> qt(c(0.025, 0.975), df=50)
+[1] -2.008559  2.008559
+. . . . . .
+> qt(c(0.025, 0.975), df=50000)
+[1] -1.960011  1.960011
+</code>
+</WRAP>
 ====== Counting the Number of Combinations ======
@@ Line 92: / Line 164: @@
 </code>
-<code>> rnorm(3, mean=c(-10,0,+10), sd=1)
+<code>> rnorm(3, mean=c(-10,0,+10), sd=1) # mean이 각 -10,0,10이고 각 mean의 sd가 1인 경우에, random score를 구할것
 [1] -11.195667   2.615493  10.294831
 </code>
@@ Line 131: / Line 204: @@
 Replacement in random sampling: Specify replace=TRUE to sample with replacement.
+<code>> set.seed(121)
+sample(world.series$year, 10)
+ [1] 1906 1963 1966 1928 1905 1924 1961 1959 1927 1934
+set.seed(121)
+sample(world.series$year, 10)
+ [1] 1906 1963 1966 1928 1905 1924 1961 1959 1927 1934
+</code>
 ====== Generating Random Sequences ======
@@ Line 197: / Line 279: @@
 </code>
-<code>> qnorm(c(0.025, 0.975))
+<code>> pnorm(73, mean=70, sd=3)
+[1] 0.8413447
+</code>
+<code>> qnorm(c(0.025, 0.975)) # 5% 바깥쪽의 점수는 약 +-2sd 점수인 -2, 2
 [1] -1.959964  1.959964
 </code>