Probabilities with pnorm function R

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Probabilities with pnorm function R

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#1 by (4 votes)

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Statistics and Probability is not my area. There was a doubt, I think I'm doing it right but the result is strange. This is my series:

y=structure(c(-0.276926746036887, 5.1002303288006, -4.45902094037891, 
-3.65240790618631, -0.554369416754141, 0.554369416754141, -2.25772100857938, 
0.375468281204183, -2.05228159135609, 3.29532926592406, -1.32662877100271, 
0.988930229485896, 7.07229348341512, -1.28656226502041, 4.2549381884453, 
3.30389447729236, 7.00232043259232, 4.7748331221784, -0.27085058060754, 
2.30352233190845, -3.22100247146746, -4.8743614652375, 2.06839209389353, 
1.89690621078412, 3.19172094544754, 2.16583767731922, -0.756218240014694, 
0.19541235523014, 3.14867268321485, -0.144151053557329, 1.01160305514936, 
1.5230454220633, 1.03840992420788, 2.71968062765501, 0.861247787660713, 
6.23477534727392, 0.5745094967259, 1.93701233271648, -4.92253766463621, 
-2.73591768775421, 4.3088310528527, 0.851783504402864, 1.72544081552388, 
-0.246815828758246, 0.166106780480357, 2.17708551212288, 0.789826760142481, 
3.52664861321393, -6.1785670816505, 3.6211660479106, 2.78553748264525, 
0.89906385457475, 2.04066324333774, 3.76850202784195, -0.0156889406364658, 
5.78634965275749, 1.26423314049121, 1.78013275057423, 3.31172073861811, 
2.4038546444218, 4.66258664624009, -13.5843132751602, -0.403750718769838, 
-3.56233196110176, -1.81026841297745, -7.46631529569132, 1.42758499302842, 
-1.24995503682966, -2.24626121938432, -3.31120681505209, 2.99957221381565, 
2.30969024677898, -2.37907446465362, -1.88443844926011, -5.12583959903528, 
5.20685002435163, -3.97367473360913, 2.67851563087302, 0.235381859142514, 
2.80824193248285, 2.89610807313333, -1.9729824987724, -3.3615385580259, 
-2.10190280870005, -2.09136151458093, -3.87094266631358, -3.2751255099671, 
1.8850792597277, 1.02942443142527, 1.36158956227371, -2.59327811511363, 
0.491897726102963, -5.78257606259328, -5.87069602591836, -3.25452200548395, 
6.39760964001113, -0.247985247978266, -6.1760320126711, 0.423565923494829, 
2.17040312533964, 7.31569072572449, -0.0871727871011863, 1.621556300047, 
-0.504340241336876, -4.50860952107559, -4.4400301974362, 2.23109247118944, 
3.28749943941721, 0.344162253040281, 3.43025943921054, -0.226072675706557, 
7.21093155995787, 1.44163507753278, -7.28040722370891, 5.74286395911455, 
-2.60392371230649, 2.31880108190266, 0.508242311922885, -2.82704339382555, 
-3.94015554915489, -0.582767936456074, 2.46930245946095, 0.553845041192957, 
-2.26755057988116, -4.00552887824704, -6.38198320512942, -1.70521028736478, 
0.982781138056799, -3.69451268069177, 0.246585308476566, 0.55308591968849, 
-1.8746351610273, 2.84001033287316, 8.83299994192822, 1.49942937302144, 
1.43144209241926, 4.82628292850097, -4.02801052546, 0.267481044687123, 
-4.19615590666126, 0.543398882231305, -1.31021176393017, -5.53706210900828, 
4.21482279511858, 1.86102992601902, 1.12800218208342, -2.20317694783819, 
-2.99809865598509, -3.9927697470259, -0.862197910255613, -3.54699081750229, 
11.1743775531587, 12.9770616075187, 3.18619731487439, 0.0317784590121939, 
1.83499639512377, 4.40415488243423, -5.97321376720503, -3.93178975523039, 
0.57591946095546, 4.42156241654894, 0.537527630812085, -0.121844321824688, 
-0.61744241735453, 0.523729365789133, 0.200706255816196, 0.519958804533455, 
-1.54404705357017, 0.86950573958392, -0.0327307895505558, -1.58683494663899, 
-0.013607190665299, -2.37306791586077, 0.440301351691902, 0.344705879693602, 
-2.62103933785511, -3.61112663327796, 4.24778609707985, 0.0313351928111716, 
-0.952260731392018, -0.158275463946334, -2.18588813267492, 0.613340421034958, 
1.41402133928985, -0.250985009120286, -0.470304013118084, 2.95224973756179, 
-1.78771862889961, -0.0758449638961822, 3.24567999848993, -1.51097540057064, 
2.86670167933881, -1.4720908190912, 0.379216944245919, 1.39885736740112, 
-2.91513301487977, -0.218721729003968, -0.174077974918679, -0.358434494209331, 
-0.262627361006695, -3.91537200150442, -3.27447875910531, -1.78701946030079, 
6.70983840143415, -1.70370125158495, -0.983681568317685, 1.51971910738222, 
-0.98330489803079, -1.75605251878265, 1.58576167333393, 1.1341537292256, 
0.533282885011355, -4.06735941384394, -0.58755861516564, 2.74957162619566, 
0.622289734338011, -0.780114417335526, -1.87837155096527, 0.226072054152004, 
2.5368406372972, -2.34466303952749, 2.09320149165367, -2.48295466772607, 
-1.39236880883822, 1.54309735856075, 1.50481181337672, -0.864243789878039, 
1.0338958791585, 2.64716085607346, -0.702852086409766, 0.341588718140706, 
0.404527399709431, 1.60509437116675, -2.76525626967899, 0.749200216946883, 
-2.27946320594848, 0.19400561795847, 3.02847564751852, -1.3395241719926, 
-0.197225968421838, 2.51515239525749, -0.895573323697879, 2.29956699052384, 
3.87915217893021, -1.07749609076405, 0.977159881194761, 3.17346515886144, 
-1.85765670111033, 1.92585418625068, 2.05443904262965, -5.10898848846342, 
2.31349795923873, 1.23878810661255, 1.69870352568162, -0.814981856773006, 
0.919933007395513, 0.693588874876505, -1.23512568593857, -1.79467292197339, 
-1.23956764741749, 1.07546772749542, 1.93739237643256, -1.99990828004241, 
2.95188431410198, -0.0635606707981018, -0.852783793313855, -2.18690287354734, 
-0.410120764875582, -4.15224123307232, -2.46163901733218, 0.957897478032321, 
-1.43716915209405, 0.446453789583057, 0.494251639553211, -3.42523616802606, 
-2.17046135806602, 4.41921336813345, -0.335763862240462, -1.32083839660705, 
-2.42762992607925, -0.82634215069915, 2.69239819821484, 1.98972704914046, 
-5.23074389240228, -2.00100680366893, -1.53953028309399, -3.55446542103525, 
-2.06954910327771, -1.95713201456629, 0.946560098399563, 3.63170421214811, 
-3.24970892660951, 1.11918342199998, -0.314074601207448, 0.840158487264464, 
-0.326895061687471, -1.40089184945676, -3.89052244917483, -0.454972014583788, 
1.63056386213887, -2.09531475147559, 1.89341455741064, -1.23150989131111, 
4.77196538893804, 1.66515332371626, 1.89683518961253, -2.35330107339518, 
1.76926250072161, 0.040693200374009, 2.10220842655976, 0.771271929555351, 
-3.61206762105296, 1.58934142527959, 0.836620342328287, -4.64587274736759, 
-2.77107914576235, 1.29533561334032, -1.04463858207546, -1.79675468861598, 
1.63218869418859, -1.94848787942505, -3.14046238852407, 0.544018624569886, 
-0.127392297256357, -0.00980613276329034, -0.365473065125499, 
-1.58733491562901, 1.01879263874898, -1.33529297306464, -1.48767751576085, 
0.980797357366159, -1.10305672543544, -1.88529499316701, 1.01498530290741, 
3.58364274627491, -0.262274385874206, 0.157049880696913, 0.0456770639576165, 
0.039702233772132, 0.271537324819782, -0.567698382468329, 0.502358961664562, 
8.51230456682989, 2.14508593839724, 9.65767019796469, 5.18629487924251, 
4.8662255853938, 1.41817523625373, 0.955339050012599, -0.167170359769503, 
-3.43010456552938, -8.79805101759148, -1.79194933762561, -1.57363442942458, 
2.41735538894081, 1.91796218652891, -2.92469310269456), .Tsp = c(1980.08333333333, 
2009.75, 12), class = "ts")

I want to calculate probabilities using the Accumulated Normal distribution function:

Prob (X

r

asked by anonymous 24.10.2016 / 22:34

1 answer

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score 4 · Accepted Answer

Yes, you are doing everything correctly.

If we are to stick to the theory, yes, these two results should be the same. If your data comes from a normal distribution, these two values must be the same.

(more generally, this result holds for any probability distribution that is symmetric to the mean)

I do not know the source of your data. I imagine they are simulated or come from some real time series. In this case, we have no guarantee that these two values will be the same. The maximum we can get are approximations for the values of these quartiles. In your case, the values were

> pnorm(quantile(y,.25), mean = mean(y), sd = sd(y), lower.tail=TRUE)
      25% 
0.2668025 
> 1-pnorm(quantile(y,.75), mean = mean(y), sd = sd(y), lower.tail=TRUE)
      75% 
0.2836341

These are very reasonable approximations for these quartiles.

See another example. Suppose I want to generate a random sample of size 100, with mean 0 and standard deviation 1, of a random variable with normal distribution. I'll call this x and estimate your average:

x <- rnorm(100, mean=0, sd=1)
mean(x)
[1] 0.0005606774

As much as you repeat this experiment, the average estimate is never going to be zero. This is very unlikely to occur. However, the approach we get is fairly good. In fact, it's so good that if I perform a hypothesis test to check if this mean is indeed zero, I get

t.test(x)

    One Sample t-test

data:  x
t = 0.00624, df = 99, p-value = 0.995
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
 -0.1777242  0.1788455
sample estimates:
   mean of x 
0.0005606774

Note that the p-value was super high, indicating that we can not reject H_0. That is, the average of this sample is, in fact, zero.

In summary, do not worry. Your logic is correct and this little difference is expected. Because of the fluctuations in the generation of this numbers, whether through simulation or their actual nature, they are not perfectly symmetrical in relation to the mean. But this feature is expected.