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--- r:probability [2017/10/30 00:26] – [Converting Probabilities to Quantiles] 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}} $
+$ 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?
-pnorm(84, mean=72, sd=15.2, lower.tail=FALSE)
+<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)
+> qnorm(.2149176, mean=72, sd=15.2, lower.tail=FALSE)
 [1] 84
-</WRAP>
+</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>
@@ Line 64: / Line 94: @@
 > 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>