{"id":491,"date":"2023-07-29T17:37:17","date_gmt":"2023-07-29T17:37:17","guid":{"rendered":"https:\/\/statorials.org\/ar\/%d8%a7%d9%84%d8%b9%d9%85%d9%84-%d9%85%d8%b9-%d8%aa%d9%88%d8%b2%d9%8a%d8%b9-%d8%a7%d9%84%d8%b7%d9%84%d8%a7%d8%a8-%d9%81%d9%8a-r-dt-qt-pt-rt\/"},"modified":"2023-07-29T17:37:17","modified_gmt":"2023-07-29T17:37:17","slug":"%d8%a7%d9%84%d8%b9%d9%85%d9%84-%d9%85%d8%b9-%d8%aa%d9%88%d8%b2%d9%8a%d8%b9-%d8%a7%d9%84%d8%b7%d9%84%d8%a7%d8%a8-%d9%81%d9%8a-r-dt-qt-pt-rt","status":"publish","type":"post","link":"https:\/\/statorials.org\/ar\/%d8%a7%d9%84%d8%b9%d9%85%d9%84-%d9%85%d8%b9-%d8%aa%d9%88%d8%b2%d9%8a%d8%b9-%d8%a7%d9%84%d8%b7%d9%84%d8%a7%d8%a8-%d9%81%d9%8a-r-dt-qt-pt-rt\/","title":{"rendered":"\u062f\u0644\u064a\u0644 \u0625\u0644\u0649 dt\u060c qt\u060c pt\u060c & rt \u0641\u064a r"},"content":{"rendered":"

\u064a\u0639\u062f \u062a\u0648\u0632\u064a\u0639 \u0627\u0644\u0637\u0627\u0644\u0628<\/strong> \u0623\u062d\u062f \u0627\u0644\u062a\u0648\u0632\u064a\u0639\u0627\u062a \u0627\u0644\u0623\u0643\u062b\u0631 \u0627\u0633\u062a\u062e\u062f\u0627\u0645\u064b\u0627 \u0641\u064a \u0627\u0644\u0625\u062d\u0635\u0627\u0621. \u064a\u0634\u0631\u062d \u0647\u0630\u0627 \u0627\u0644\u0628\u0631\u0646\u0627\u0645\u062c \u0627\u0644\u062a\u0639\u0644\u064a\u0645\u064a \u0643\u064a\u0641\u064a\u0629 \u0627\u0644\u0639\u0645\u0644 \u0645\u0639 \u062a\u0648\u0632\u064a\u0639 Student t \u0641\u064a \u0644\u063a\u0629 R \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0627\u0644\u062f\u0648\u0627\u0644 dt()<\/strong> \u0648 qt()<\/strong> \u0648 pt()<\/strong> \u0648 rt()<\/strong> .<\/span><\/p>\n

dt<\/span><\/strong><\/h2>\n

\u062a\u0642\u0648\u0645 \u0627\u0644\u062f\u0627\u0644\u0629 dt<\/b> \u0628\u0625\u0631\u062c\u0627\u0639 \u0642\u064a\u0645\u0629 \u062f\u0627\u0644\u0629 \u0627\u0644\u0643\u062b\u0627\u0641\u0629 \u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644\u064a\u0629 (pdf) \u0644\u062a\u0648\u0632\u064a\u0639 \u0627\u0644\u0637\u0627\u0644\u0628 t \u0641\u064a \u0636\u0648\u0621 \u0645\u062a\u063a\u064a\u0631 \u0639\u0634\u0648\u0627\u0626\u064a \u0645\u0639\u064a\u0646 x<\/em> \u0648\u062f\u0631\u062c\u0627\u062a \u0627\u0644\u062d\u0631\u064a\u0629 df<\/em> . \u0628\u0646\u0627\u0621 \u0627\u0644\u062c\u0645\u0644\u0629 \u0644\u0627\u0633\u062a\u062e\u062f\u0627\u0645 dt \u0647\u0648 \u0643\u0645\u0627 \u064a\u0644\u064a:<\/span><\/p>\n

\u062f\u064a\u0646\u0627\u0631\u0627 (\u0633\u060c \u0645\u062f\u0627\u0641\u0639)<\/span><\/strong><\/p>\n

\u064a\u0648\u0636\u062d \u0627\u0644\u062a\u0639\u0644\u064a\u0645\u0629 \u0627\u0644\u0628\u0631\u0645\u062c\u064a\u0629 \u0627\u0644\u062a\u0627\u0644\u064a\u0629 \u0628\u0639\u0636 \u0627\u0644\u0623\u0645\u062b\u0644\u0629 \u0639\u0644\u0649 dt<\/b> \u0623\u062b\u0646\u0627\u0621 \u0627\u0644\u0639\u0645\u0644:<\/span><\/p>\n

 #find the value of the Student t distribution pdf at x = 0 with 20 degrees of freedom<\/span>\ndt(x = 0, df = 20)\n\n#[1] 0.3939886\n\n#by default, R assumes the first argument is x<\/em> and the second argument is df<\/em><\/span>\ndt(0, 20)\n\n#[1] 0.3939886\n#find the value of the Student t distribution pdf at x = 1 with 30 degrees of freedom\ndt(1, 30)\n<\/span>\n#[1] 0.2379933\n<\/span><\/span><\/strong><\/pre>\n
 \u0639\u0627\u062f\u0629\u064b\u060c \u0639\u0646\u062f \u0645\u062d\u0627\u0648\u0644\u0629 \u062d\u0644 \u0623\u0633\u0626\u0644\u0629 \u062d\u0648\u0644 \u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644\u064a\u0629 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u062a\u0648\u0632\u064a\u0639 \u0627\u0644\u0637\u0627\u0644\u0628\u060c \u063a\u0627\u0644\u0628\u064b\u0627 \u0645\u0627 \u062a\u0633\u062a\u062e\u062f\u0645 pt<\/b> \u0628\u062f\u0644\u0627\u064b \u0645\u0646 dt<\/strong> . \u0623\u062d\u062f \u0627\u0644\u062a\u0637\u0628\u064a\u0642\u0627\u062a \u0627\u0644\u0645\u0641\u064a\u062f\u0629 \u0644\u0640 dt<\/strong> \u0647\u0648 \u0625\u0646\u0634\u0627\u0621 \u0645\u062e\u0637\u0637 \u062a\u0648\u0632\u064a\u0639 Student t \u0641\u064a R. \u064a\u0648\u0636\u062d \u0627\u0644\u0643\u0648\u062f \u0627\u0644\u062a\u0627\u0644\u064a \u0643\u064a\u0641\u064a\u0629 \u0627\u0644\u0642\u064a\u0627\u0645 \u0628\u0630\u0644\u0643:<\/span><\/p>\n
 #Create a sequence of 100 equally spaced numbers between -4 and 4<\/span>\nx <- seq(-4, 4, length=100)\n\n#create a vector of values that shows the height of the probability distribution<\/span>\n#for each value in x, using 20 degrees of freedom<\/span>\ny <- dt(x = x, df = 20)\n\n#plot x and y as a scatterplot with connected lines (type = \"l\") and add<\/span>\n#an x-axis with custom labels<\/span>\nplot(x,y, type = \"l\", lwd = 2, axes = FALSE, xlab = \"\", ylab = \"\")\naxis(1, at = -3:3, labels = c(\"-3s\", \"-2s\", \"-1s\", \"mean\", \"1s\", \"2s\", \"3s\"))<\/strong><\/pre>\n
 \u064a\u0624\u062f\u064a \u0647\u0630\u0627 \u0625\u0644\u0649 \u0625\u0646\u0634\u0627\u0621 \u0627\u0644\u0645\u0624\u0627\u0645\u0631\u0629 \u0627\u0644\u062a\u0627\u0644\u064a\u0629:<\/span><\/p>\n
 \u0646\u0642\u0637\u0629<\/span><\/strong><\/h2>\n
 \u062a\u0631\u062c\u0639 \u0627\u0644\u062f\u0627\u0644\u0629 pt<\/b> \u0642\u064a\u0645\u0629 \u062f\u0627\u0644\u0629 \u0627\u0644\u0643\u062b\u0627\u0641\u0629 \u0627\u0644\u062a\u0631\u0627\u0643\u0645\u064a\u0629 (cdf) \u0644\u062a\u0648\u0632\u064a\u0639 \u0627\u0644\u0637\u0627\u0644\u0628 t \u0641\u064a \u0636\u0648\u0621 \u0645\u062a\u063a\u064a\u0631 \u0639\u0634\u0648\u0627\u0626\u064a \u0645\u0639\u064a\u0646 x<\/em> \u0648\u062f\u0631\u062c\u0627\u062a \u0627\u0644\u062d\u0631\u064a\u0629 df<\/em> . \u0628\u0646\u0627\u0621 \u0627\u0644\u062c\u0645\u0644\u0629 \u0644\u0627\u0633\u062a\u062e\u062f\u0627\u0645 pnorm \u0647\u0648 \u0643\u0645\u0627 \u064a\u0644\u064a:<\/span><\/p>\n
 \u062d\u0632\u0628 \u0627\u0644\u0639\u0645\u0627\u0644 (\u0633\u060c \u0645\u062f\u0627\u0641\u0639)<\/span><\/strong><\/p>\n
 \u0628\u0639\u0628\u0627\u0631\u0627\u062a \u0628\u0633\u064a\u0637\u0629\u060c \u062a\u0642\u0648\u0645 pt<\/b> \u0628\u0625\u0631\u062c\u0627\u0639 \u0627\u0644\u0645\u0646\u0637\u0642\u0629 \u0627\u0644\u0645\u0648\u062c\u0648\u062f\u0629 \u0639\u0644\u0649 \u064a\u0633\u0627\u0631 \u0642\u064a\u0645\u0629 x<\/em> \u0645\u0639\u064a\u0646\u0629 \u0641\u064a \u062a\u0648\u0632\u064a\u0639 t \u0627\u0644\u062e\u0627\u0635 \u0628\u0627\u0644\u0637\u0627\u0644\u0628. \u0625\u0630\u0627 \u0643\u0646\u062a \u0645\u0647\u062a\u0645\u064b\u0627 \u0628\u0627\u0644\u0645\u0646\u0637\u0642\u0629 \u0627\u0644\u0645\u0648\u062c\u0648\u062f\u0629 \u0639\u0644\u0649 \u064a\u0645\u064a\u0646 \u0642\u064a\u0645\u0629 x<\/em> \u0645\u0639\u064a\u0646\u0629\u060c \u0641\u064a\u0645\u0643\u0646\u0643 \u0628\u0628\u0633\u0627\u0637\u0629 \u0625\u0636\u0627\u0641\u0629 \u0627\u0644\u0648\u0633\u064a\u0637\u0629 Lower.tail = FALSE<\/strong><\/span><\/p>\n
 \u062d\u0632\u0628 \u0627\u0644\u0639\u0645\u0627\u0644 (\u0633\u060c \u0645\u062f\u0627\u0641\u0639\u060c Lower.tail = FALSE)<\/span><\/strong><\/p>\n
 \u062a\u0648\u0636\u062d \u0627\u0644\u0623\u0645\u062b\u0644\u0629 \u0627\u0644\u062a\u0627\u0644\u064a\u0629 \u0643\u064a\u0641\u064a\u0629 \u062d\u0644 \u0628\u0639\u0636 \u0627\u0644\u0623\u0633\u0626\u0644\u0629 \u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644\u064a\u0629 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 pt.<\/span><\/p>\n
 \u0645\u062b\u0627\u0644 1:<\/strong> \u0623\u0648\u062c\u062f \u0627\u0644\u0645\u0633\u0627\u062d\u0629 \u0627\u0644\u0645\u0648\u062c\u0648\u062f\u0629 \u0639\u0644\u0649 \u064a\u0633\u0627\u0631<\/strong> \u0625\u062d\u0635\u0627\u0626\u064a\u0629 t \u0628\u0642\u064a\u0645\u0629 -0.785 \u064814 \u062f\u0631\u062c\u0629 \u062d\u0631\u064a\u0629.<\/em><\/span><\/p>\n
 pt(-0.785, 14)\n\n#[1] 0.2227675\n<\/span><\/span><\/strong><\/pre>\n
 \u0645\u062b\u0627\u0644 2:<\/strong> \u0623\u0648\u062c\u062f \u0627\u0644\u0645\u0633\u0627\u062d\u0629 \u0627\u0644\u0648\u0627\u0642\u0639\u0629 \u0639\u0644\u0649 \u064a\u0645\u064a\u0646<\/strong> \u0625\u062d\u0635\u0627\u0626\u064a\u0629 t \u0628\u0642\u064a\u0645\u0629 -0.785 \u064814 \u062f\u0631\u062c\u0629 \u062d\u0631\u064a\u0629.<\/em><\/span><\/p>\n
 #the following approaches produce equivalent results\n\n#1 - area to the left<\/span>\n1 - pt(-0.785, 14)\n\n#[1] 0.7772325\n\n#area to the right<\/span>\npt(-0.785, 14, lower.tail = FALSE)\n\n#[1] 0.7772325 \n<\/span><\/span><\/span><\/strong><\/pre>\n
 \u0645\u062b\u0627\u0644 3:<\/strong> \u0623\u0648\u062c\u062f \u0627\u0644\u0645\u0633\u0627\u062d\u0629 \u0627\u0644\u0625\u062c\u0645\u0627\u0644\u064a\u0629 \u0641\u064a \u062a\u0648\u0632\u064a\u0639 t \u0644\u0644\u0637\u0627\u0644\u0628 \u0645\u0639 14 \u062f\u0631\u062c\u0629 \u062d\u0631\u064a\u0629 \u062a\u0642\u0639 \u0639\u0644\u0649 \u064a\u0633\u0627\u0631 -0.785 \u0623\u0648 \u0639\u0644\u0649 \u064a\u0645\u064a\u0646 0.785.<\/em><\/span><\/p>\n
 pt (-0.785, 14)<\/span><\/span> + pt (0.785, 14, lower.tail = FALSE)<\/span><\/span>\n\n#[1] 0.4455351<\/strong><\/pre>\n
 \u0643\u064a\u0648 \u062a\u064a<\/span><\/strong><\/h2>\n
 \u062a\u0642\u0648\u0645 \u0627\u0644\u062f\u0627\u0644\u0629 qt<\/b> \u0628\u0625\u0631\u062c\u0627\u0639 \u0642\u064a\u0645\u0629 \u062f\u0627\u0644\u0629 \u0627\u0644\u0643\u062b\u0627\u0641\u0629 \u0627\u0644\u062a\u0631\u0627\u0643\u0645\u064a\u0629 \u0627\u0644\u0639\u0643\u0633\u064a\u0629 (cdf) \u0644\u062a\u0648\u0632\u064a\u0639 \u0627\u0644\u0637\u0627\u0644\u0628 t \u0628\u0627\u0644\u0646\u0638\u0631 \u0625\u0644\u0649 \u0645\u062a\u063a\u064a\u0631 \u0639\u0634\u0648\u0627\u0626\u064a \u0645\u0639\u064a\u0646 x<\/em> \u0648\u062f\u0631\u062c\u0627\u062a \u0627\u0644\u062d\u0631\u064a\u0629 df.<\/em> \u0628\u0646\u0627\u0621 \u0627\u0644\u062c\u0645\u0644\u0629 \u0644\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0643\u064a\u0648 \u062a\u064a \u0647\u0648 \u0643\u0645\u0627 \u064a\u0644\u064a:<\/span><\/p>\n
 \u0643\u064a\u0648 \u062a\u064a (\u0633\u060c \u0645\u062f\u0627\u0641\u0639)<\/span><\/strong><\/p>\n
 \u0628\u0639\u0628\u0627\u0631\u0627\u062a \u0628\u0633\u064a\u0637\u0629\u060c \u064a\u0645\u0643\u0646\u0643 \u0627\u0633\u062a\u062e\u062f\u0627\u0645 qt<\/strong> \u0644\u0645\u0639\u0631\u0641\u0629 \u062f\u0631\u062c\u0629 t \u0644\u0644\u0643\u0645 pth<\/sup> \u0644\u062a\u0648\u0632\u064a\u0639 t \u0644\u0644\u0637\u0627\u0644\u0628.<\/span><\/p>\n
 \u064a\u0648\u0636\u062d \u0627\u0644\u062a\u0639\u0644\u064a\u0645\u0629 \u0627\u0644\u0628\u0631\u0645\u062c\u064a\u0629 \u0627\u0644\u062a\u0627\u0644\u064a\u0629 \u0628\u0639\u0636 \u0627\u0644\u0623\u0645\u062b\u0644\u0629 \u0639\u0644\u0649 \u0643\u064a\u0648 \u062a\u064a<\/b> \u0641\u064a \u0627\u0644\u0639\u0645\u0644:<\/span><\/p>\n
 #find the t-score of the 99th quantile of the Student t distribution with df = 20\n<\/span>qt(.99, df = 20)\n\n#[1][1]2.527977\n\n#find the t-score of the 95th quantile of the Student t distribution with df = 20<\/span>\nqt(.95, df = 20)\n\n#[1]1.724718\n\n#find the t-score of the 90th quantile of the Student t distribution with df = 20<\/span>\nqt(.9, df = 20)\n\n#[1]1.325341<\/span>\n<\/span><\/strong><\/pre>\n
 \u0644\u0627\u062d\u0638 \u0623\u0646 \u0627\u0644\u0642\u064a\u0645 \u0627\u0644\u062d\u0631\u062c\u0629 \u0627\u0644\u062a\u064a \u0648\u062c\u062f\u062a\u0647\u0627 qt<\/strong> \u0633\u062a\u062a\u0648\u0627\u0641\u0642 \u0645\u0639 \u0627\u0644\u0642\u064a\u0645 \u0627\u0644\u062d\u0631\u062c\u0629 \u0627\u0644\u0645\u0648\u062c\u0648\u062f\u0629 \u0641\u064a \u062c\u062f\u0648\u0644 \u062a\u0648\u0632\u064a\u0639 t \u0648\u0643\u0630\u0644\u0643 \u0627\u0644\u0642\u064a\u0645 \u0627\u0644\u062d\u0631\u062c\u0629 \u0627\u0644\u062a\u064a \u064a\u0645\u0643\u0646 \u0627\u0644\u0639\u062b\u0648\u0631 \u0639\u0644\u064a\u0647\u0627 \u0628\u0648\u0627\u0633\u0637\u0629 \u062d\u0627\u0633\u0628\u0629 \u062a\u0648\u0632\u064a\u0639 t \u0627\u0644\u0639\u0643\u0633\u064a\u0629.<\/span><\/p>\n
 \u063a<\/span><\/strong><\/h2>\n
 \u062a\u0642\u0648\u0645 \u0627\u0644\u062f\u0627\u0644\u0629 rt<\/b> \u0628\u0625\u0646\u0634\u0627\u0621 \u0645\u062a\u062c\u0647 \u0644\u0644\u0645\u062a\u063a\u064a\u0631\u0627\u062a \u0627\u0644\u0639\u0634\u0648\u0627\u0626\u064a\u0629 \u0627\u0644\u062a\u064a \u062a\u062a\u0628\u0639 \u062a\u0648\u0632\u064a\u0639 \u0627\u0644\u0637\u0627\u0644\u0628 t \u0645\u0639 \u0625\u0639\u0637\u0627\u0621 \u0637\u0648\u0644 \u0627\u0644\u0645\u062a\u062c\u0647 n<\/em> \u0648\u062f\u0631\u062c\u0627\u062a \u0627\u0644\u062d\u0631\u064a\u0629 df<\/em> . \u0628\u0646\u0627\u0621 \u0627\u0644\u062c\u0645\u0644\u0629 \u0644\u0627\u0633\u062a\u062e\u062f\u0627\u0645 rt \u0647\u0648 \u0643\u0645\u0627 \u064a\u0644\u064a:<\/span><\/p>\n
 \u063a (\u0646\u060c \u0645\u062f\u0627\u0641\u0639)<\/span><\/strong><\/p>\n
 \u064a\u0648\u0636\u062d \u0627\u0644\u062a\u0639\u0644\u064a\u0645\u0629 \u0627\u0644\u0628\u0631\u0645\u062c\u064a\u0629 \u0627\u0644\u062a\u0627\u0644\u064a\u0629 \u0628\u0639\u0636 \u0627\u0644\u0623\u0645\u062b\u0644\u0629 \u0639\u0644\u0649 RT<\/b> \u0641\u064a \u0627\u0644\u0639\u0645\u0644:<\/span><\/p>\n
 #generate a vector of 5 random variables that follows a Student t distribution\n#with df = 20<\/span>\nrt(n = 5, df = 20)\n\n#[1] -1.7422445 0.9560782 0.6635823 1.2122289 -0.7052825\n\n#generate a vector of 1000 random variables that follows a Student t distribution\n#with df = 40\nnarrowDistribution <- rt(1000, 40)\n\n#generate a vector of 1000 random variables that follows a Student t distribution\n#with df = 5\n<\/span>wideDistribution <- rt(1000, 5)\n\n#generate two histograms to view these two distributions side by side, and specify\n#50 bars in histogram,<\/span>\npar(mfrow=c(1, 2)) #one row, two columns\nhist(narrowDistribution, breaks=50, xlim = c(-6, 4)) \nhist(wideDistribution, breaks=50, xlim = c(-6, 4))\n<\/span><\/span><\/strong><\/pre>\n
 \u064a\u0624\u062f\u064a \u0647\u0630\u0627 \u0625\u0644\u0649 \u0625\u0646\u0634\u0627\u0621 \u0627\u0644\u0631\u0633\u0648\u0645 \u0627\u0644\u0628\u064a\u0627\u0646\u064a\u0629 \u0627\u0644\u062a\u0627\u0644\u064a\u0629:<\/span><\/p>\n
 \u0644\u0627\u062d\u0638 \u0643\u064a\u0641 \u0623\u0646 \u0627\u0644\u062a\u0648\u0632\u064a\u0639 \u0627\u0644\u0648\u0627\u0633\u0639 \u0623\u0648\u0633\u0639 \u0645\u0646 \u0627\u0644\u062a\u0648\u0632\u064a\u0639 \u0627\u0644\u0636\u064a\u0642. \u0648\u0628\u0627\u0644\u0641\u0639\u0644 \u0641\u0642\u062f \u062d\u062f\u062f\u0646\u0627 \u0623\u0646 \u062f\u0631\u062c\u0627\u062a \u0627\u0644\u062d\u0631\u064a\u0629 \u0641\u064a \u0627\u0644\u062a\u0648\u0632\u064a\u0639 \u0627\u0644\u0648\u0627\u0633\u0639 \u0643\u0627\u0646\u062a 5 \u0645\u0642\u0627\u0628\u0644 40 \u0641\u064a \u0627\u0644\u062a\u0648\u0632\u064a\u0639 \u0627\u0644\u0636\u064a\u0642. \u0643\u0644\u0645\u0627 \u0642\u0644\u062a \u062f\u0631\u062c\u0627\u062a \u0627\u0644\u062d\u0631\u064a\u0629\u060c \u0643\u0644\u0645\u0627 \u0643\u0627\u0646 \u062a\u0648\u0632\u064a\u0639 \u0627\u0644\u0637\u0627\u0644\u0628 \u0623\u0648\u0633\u0639.<\/span><\/p>\n
 \u0642\u0631\u0627\u0621\u0629 \u0645\u062a\u0639\u0645\u0642\u0629:
 \u062f\u0644\u064a\u0644 \u0625\u0644\u0649 dnorm \u0648pnorm \u0648qnorm \u0648rnorm \u0641\u064a R
 \u062f\u0644\u064a\u0644 \u0625\u0644\u0649 dbinom \u0648pbinom \u0648qbinom \u0648rbinom \u0641\u064a R
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\u064a\u0639\u062f \u062a\u0648\u0632\u064a\u0639 \u0627\u0644\u0637\u0627\u0644\u0628 \u0623\u062d\u062f \u0627\u0644\u062a\u0648\u0632\u064a\u0639\u0627\u062a \u0627\u0644\u0623\u0643\u062b\u0631 \u0627\u0633\u062a\u062e\u062f\u0627\u0645\u064b\u0627 \u0641\u064a \u0627\u0644\u0625\u062d\u0635\u0627\u0621. \u064a\u0634\u0631\u062d \u0647\u0630\u0627 \u0627\u0644\u0628\u0631\u0646\u0627\u0645\u062c \u0627\u0644\u062a\u0639\u0644\u064a\u0645\u064a \u0643\u064a\u0641\u064a\u0629 \u0627\u0644\u0639\u0645\u0644 \u0645\u0639 \u062a\u0648\u0632\u064a\u0639 Student t \u0641\u064a \u0644\u063a\u0629 R \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0627\u0644\u062f\u0648\u0627\u0644 dt() \u0648 qt() \u0648 pt() \u0648 rt() . dt \u062a\u0642\u0648\u0645 \u0627\u0644\u062f\u0627\u0644\u0629 dt \u0628\u0625\u0631\u062c\u0627\u0639 \u0642\u064a\u0645\u0629 \u062f\u0627\u0644\u0629 \u0627\u0644\u0643\u062b\u0627\u0641\u0629 \u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644\u064a\u0629 (pdf) \u0644\u062a\u0648\u0632\u064a\u0639 \u0627\u0644\u0637\u0627\u0644\u0628 t \u0641\u064a \u0636\u0648\u0621 \u0645\u062a\u063a\u064a\u0631 \u0639\u0634\u0648\u0627\u0626\u064a \u0645\u0639\u064a\u0646 x \u0648\u062f\u0631\u062c\u0627\u062a \u0627\u0644\u062d\u0631\u064a\u0629 df . 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