install.packages("cubature")
library(cubature)

beta_moment <- function(a, k) {
  f <- function(t) t^(a - 1) * (log(1 - t))^k
  integrate(f, lower = 0, upper = 1)$value
}

bfgmtlee_moment_exact <- function(r, s, alpha1, lambda1, alpha2, lambda2, delta) {
  const <- (-1)^(r + s) / (2^(r + s) * lambda1^r * lambda2^s)
  B1_r  <- beta_moment(alpha1, r)
  B2_s  <- beta_moment(alpha2, s)
  B1_2r <- beta_moment(2 * alpha1, r)
  B2_2s <- beta_moment(2 * alpha2, s)
  term1 <- B1_r * B2_s
  term2 <- (B1_r - 2 * B1_2r) * (B2_s - 2 * B2_2s)
  const * alpha1 * alpha2 * (term1 + delta * term2)
}

delta  <- 0.6
alpha1 <- 3
alpha2 <- 5

r_vals <- c(1, 2, 3, 5)
s_vals <- c(1, 2, 3, 5)

# حالت اول
lambda1 <- 0.7; lambda2 <- 0.9

moment_mat <- matrix(0, nrow = length(r_vals), ncol = length(s_vals))
rownames(moment_mat) <- paste0("r=", r_vals)
colnames(moment_mat) <- paste0("s=", s_vals)

for (i in 1:length(r_vals)) {
  for (j in 1:length(s_vals)) {
    moment_mat[i, j] <- bfgmtlee_moment_exact(r_vals[i], s_vals[j],
                                              alpha1, lambda1, alpha2, lambda2, delta)
  }
}
cat("\n--- Lambda (moderate): δ=0.6, α1=3, λ1=0.7, α2=5, λ2=0.9 ---\n")
print(round(moment_mat, 3))
M_mod <- moment_mat

# حالت دوم
lambda1 <- 1.2; lambda2 <- 1.4

moment_mat <- matrix(0, nrow = length(r_vals), ncol = length(s_vals))
rownames(moment_mat) <- paste0("r=", r_vals)
colnames(moment_mat) <- paste0("s=", s_vals)

for (i in 1:length(r_vals)) {
  for (j in 1:length(s_vals)) {
    moment_mat[i, j] <- bfgmtlee_moment_exact(r_vals[i], s_vals[j],
                                              alpha1, lambda1, alpha2, lambda2, delta)
  }
}
cat("\n--- Lambda (larger): δ=0.6, α1=3, λ1=1.2, α2=5, λ2=1.4 ---\n")
print(round(moment_mat, 3))
M_big <- moment_mat<pre>An Error occurred while handling another error:
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Stack trace:
#0 /var/www/html/prof-homepages/vendor/yiisoft/yii2/web/Response.php(339): yii\web\Response-&gt;sendHeaders()
#1 /var/www/html/prof-homepages/vendor/yiisoft/yii2/web/ErrorHandler.php(136): yii\web\Response-&gt;send()
#2 /var/www/html/prof-homepages/vendor/yiisoft/yii2/base/ErrorHandler.php(135): yii\web\ErrorHandler-&gt;renderException()
#3 [internal function]: yii\base\ErrorHandler-&gt;handleException()
#4 {main}
Previous exception:
yii\web\HeadersAlreadySentException: Headers already sent in  on line 0. in /var/www/html/prof-homepages/vendor/yiisoft/yii2/web/Response.php:366
Stack trace:
#0 /var/www/html/prof-homepages/vendor/yiisoft/yii2/web/Response.php(339): yii\web\Response-&gt;sendHeaders()
#1 /var/www/html/prof-homepages/vendor/yiisoft/yii2/base/Application.php(656): yii\web\Response-&gt;send()
#2 /var/www/html/prof-homepages/vendor/faravaghi/yii2-filemanager/models/Files.php(696): yii\base\Application-&gt;end()
#3 /var/www/html/prof-homepages/vendor/faravaghi/yii2-filemanager/controllers/FilesController.php(484): faravaghi\filemanager\models\Files-&gt;getFile()
#4 [internal function]: faravaghi\filemanager\controllers\FilesController-&gt;actionGetFile()
#5 /var/www/html/prof-homepages/vendor/yiisoft/yii2/base/InlineAction.php(57): call_user_func_array()
#6 /var/www/html/prof-homepages/vendor/yiisoft/yii2/base/Controller.php(180): yii\base\InlineAction-&gt;runWithParams()
#7 /var/www/html/prof-homepages/vendor/yiisoft/yii2/base/Module.php(528): yii\base\Controller-&gt;runAction()
#8 /var/www/html/prof-homepages/vendor/yiisoft/yii2/web/Application.php(103): yii\base\Module-&gt;runAction()
#9 /var/www/html/prof-homepages/vendor/yiisoft/yii2/base/Application.php(386): yii\web\Application-&gt;handleRequest()
#10 /var/www/html/prof-homepages/backend/web/index.php(16): yii\base\Application-&gt;run()
#11 {main}</pre>