From 5323793daa2c9ee51fa52eae3182becdd7accb2c Mon Sep 17 00:00:00 2001
From: Arthur DANJOU <arthurdanjou@outlook.fr>
Date: Wed, 13 Nov 2024 18:30:34 +0100
Subject: [PATCH] Add Exo 10-11-12

---
 M1/Monte Carlo Methods/Exercise10.rmd | 10 +-----
 M1/Monte Carlo Methods/Exercise12.rmd | 48 +++++++++++++++++++++++++++
 M1/Monte Carlo Methods/Exercise13.rmd |  6 ++++
 3 files changed, 55 insertions(+), 9 deletions(-)
 create mode 100644 M1/Monte Carlo Methods/Exercise12.rmd
 create mode 100644 M1/Monte Carlo Methods/Exercise13.rmd

diff --git a/M1/Monte Carlo Methods/Exercise10.rmd b/M1/Monte Carlo Methods/Exercise10.rmd
index 2be4a54..5bff6d7 100644
--- a/M1/Monte Carlo Methods/Exercise10.rmd	
+++ b/M1/Monte Carlo Methods/Exercise10.rmd	
@@ -37,12 +37,4 @@ h <- function(x, y) {
 
 I2 <- mean(h(X, Y))
 I2
-```
-
-### Exponential + Truncated Normal distributions
-
-```{r}
-
-```
-
-## Computational cost of these methods.
\ No newline at end of file
+```
\ No newline at end of file
diff --git a/M1/Monte Carlo Methods/Exercise12.rmd b/M1/Monte Carlo Methods/Exercise12.rmd
new file mode 100644
index 0000000..4dcfd09
--- /dev/null
+++ b/M1/Monte Carlo Methods/Exercise12.rmd	
@@ -0,0 +1,48 @@
+# Exercise 12 : Rejection vs Importance Sampling
+```{r}
+set.seed(123)
+n <- 10000
+
+f <- function(x, y) {
+  3 / 2 * exp(-3 / 2 * x) * (x > 0) * 1 / (2 * sqrt(pi)) *
+    exp(-y^2 / 4) *
+    (x > 0)
+}
+
+g <- function(x, y, lambda) {
+  (3 / 2 * exp(-3 / 2 * x)) *
+    (x >= 0) *
+    (lambda * exp(-lambda * (y - 2))) *
+    (y >= 2)
+}
+
+h <- function(x, y) {
+  sqrt(x + y) *
+    sin(y^4) *
+    (x <= 5) *
+    (x > 0) *
+    (y >= 2) *
+    (4 * sqrt(pi) / 3)
+}
+
+X <- rexp(n, 3 / 2)
+Y <- rexp(n, 2) + 2
+
+mean(h(X, Y) * f(X, Y) / g(X, Y, 0.4))
+```
+
+### Monte Carlo method
+
+```{r}
+set.seed(123)
+n <- 10e4
+
+X <- rexp(n, 3 / 2)
+Y <- rexp(n, 3 / 2) + 2
+
+X[X > 5] <- 0
+X[X < 0] <- 0
+Y[Y < 2] <- 0
+
+mean(h(X, Y))
+```
\ No newline at end of file
diff --git a/M1/Monte Carlo Methods/Exercise13.rmd b/M1/Monte Carlo Methods/Exercise13.rmd
new file mode 100644
index 0000000..71ac457
--- /dev/null
+++ b/M1/Monte Carlo Methods/Exercise13.rmd	
@@ -0,0 +1,6 @@
+# Exercise 13 : Gaussian integral and Variance reduction
+
+```{r}
+n <- 10000
+
+```
\ No newline at end of file