pushing 2b

Oblig/2b/2b.R

@@ -0,0 +1,609 @@
+#Oppgave 40 i læreboka
+#a)
+P_knepet = 1/4
+P_knepet
+
+
+#b)
+P_2av5 = dbinom(2, 5, 1/4)
+P_2av5
+
+
+#c)
+ganger_tatt_totalt = 2
+antall_passeringer = ganger_tatt_totalt / P_knepet
+standardavvik = sqrt(antall_passeringer * P_knepet * (1 - P_knepet))
+
+antall_passeringer
+standardavvik
+
+
+#d)
+passeringer = 12
+leser_ved_passering = 7
+antall_tatt = 4
+ønsket_tatt_og_leser = 3
+P_leste_og_tatt = dhyper(ønsket_tatt_og_leser, leser_ved_passering, 
+                        ( passeringer - leser_ved_passering ), antall_tatt)
+P_leste_og_tatt
+
+
+#f)
+antall_biler = 16384
+P_wunderbaum = 1 / 1024
+lambda = antall_biler * P_wunderbaum
+
+P_nøyaktig_15 = dpois(15, lambda)
+P_nøyaktig_15
+
+
+#2a)
+rbinom(1, 10, 2/3)
+rbinom(10, 1, 2/3)
+sum(rbinom(10, 1, 2/3))
+
+
+#b) 
+data = c()
+
+for (i in 1:50) {
+  k = rbinom(1, 10, 2/3)
+  data = c(data, k)
+}
+
+table(data)
+hist(data, breaks = 100, col = "purple")
+
+#d)
+for (j in 1:5) {
+  data = c()
+  
+  for (i in 1:50) {
+    k = rbinom(1, 10, 2/3)
+    data = c(data, k)
+  }
+  
+  hist(data, breaks = 100, col = "purple",
+       main = paste("Histogram, kjøring", j),
+       xlab = "Antall suksesser")
+}
+
+
+#e)
+for (j in 1:5) {
+  data = c()
+  
+  for (i in 1:100000) {
+    k = rbinom(1, 10, 2/3)
+    data = c(data, k)
+  }
+  
+  hist(data, breaks = 100, col = "purple",
+       main = paste("Histogram, 100000 simuleringer, kjøring", j),
+       xlab = "Antall suksesser")
+}
+
+
+#f)
+x = 0:10
+y = dbinom(x, 10, 2/3)
+
+plot(x, y, type = "h", lwd = 3, col = "maroon",
+     main = "Bin(10, 2/3)",
+     xlab = "Antall suksesser", ylab = "P(X = x)")
+points(x, y, pch = 19, col = "maroon")
+
+
+
+#3)
+for (j in 1:5) {
+  data = c()
+  
+  for (i in 1:200) {
+    k = rpois(1, 2)
+    data = c(data, k)
+  }
+  
+  hist(data,
+       breaks = seq(-0.5, max(data) + 0.5, 1),
+       col = "purple",
+       main = paste("Poisson(2), kjøring", j),
+       xlab = "Antall hendelser")
+}
+
+
+
+x = 0:10
+y = dpois(x, 2)
+
+plot(x, y, type = "h", lwd = 3, col = "maroon",
+     main = "Poisson(2)",
+     xlab = "Antall hendelser", ylab = "P(X = x)")
+points(x, y, pch = 19, col = "maroon")
+
+
+
+data = c()
+
+for (i in 1:100000) {
+  k = rpois(1, 2)
+  data = c(data, k)
+}
+
+hist(data,
+     breaks = seq(-0.5, max(data) + 0.5, 1),
+     col = "purple",
+     main = "Poisson(2), 100000 simuleringer",
+     xlab = "Antall hendelser")
+
+
+
+par(mfrow = c(1, 1))
+
+# 4a
+png("/Users/chris/Maths/MA-223/Oblig/2b/oppg4a.png", width = 1400, height = 700)
+par(mfrow = c(1, 2))
+
+mu = 37
+sigma = 5
+
+# Sannsynlighet
+P_X_ge_33 = pnorm(33, mean = mu, sd = sigma, lower.tail = FALSE)
+P_X_ge_33
+
+# PDF med skyggelegging
+xVals = seq(15, 60, 0.01)
+yVals = dnorm(xVals, mean = mu, sd = sigma)
+
+plot(xVals, yVals, type = "l", lwd = 2,
+     main = "PDF for N(37,5)",
+     xlab = "x", ylab = "f(x)")
+
+shade_x = seq(33, 60, 0.01)
+shade_y = dnorm(shade_x, mean = mu, sd = sigma)
+
+polygon(c(33, shade_x, 60),
+        c(0, shade_y, 0),
+        col = "plum", border = NA)
+
+lines(xVals, yVals, lwd = 2)
+abline(v = 33, col = "maroon", lwd = 2)
+
+# CDF med markering
+yCumVals = pnorm(xVals, mean = mu, sd = sigma)
+
+plot(xVals, yCumVals, type = "l", lwd = 2,
+     main = "CDF for N(37,5)",
+     xlab = "x", ylab = "F(x)")
+
+F_33 = pnorm(33, mean = mu, sd = sigma)
+
+points(33, F_33, pch = 19, col = "maroon")
+abline(v = 33, col = "maroon", lty = 2)
+abline(h = F_33, col = "maroon", lty = 2)
+
+dev.off()
+
+
+# 4b
+png("/Users/chris/Maths/MA-223/Oblig/2b/oppg4b.png", width = 1400, height = 700)
+par(mfrow = c(1, 2))
+
+mu = 2
+sigma = 1
+
+a = qnorm(0.05, mean = mu, sd = sigma)
+a
+
+# PDF
+xVals = seq(-2, 6, 0.01)
+yVals = dnorm(xVals, mean = mu, sd = sigma)
+
+plot(xVals, yVals, type = "l", lwd = 2,
+     main = "PDF for N(2,1)",
+     xlab = "x", ylab = "f(x)")
+
+shade_x = seq(-2, a, 0.01)
+shade_y = dnorm(shade_x, mean = mu, sd = sigma)
+
+polygon(c(-2, shade_x, a),
+        c(0, shade_y, 0),
+        col = "plum", border = NA)
+
+lines(xVals, yVals, lwd = 2)
+abline(v = a, col = "maroon", lwd = 2)
+
+# CDF
+yCumVals = pnorm(xVals, mean = mu, sd = sigma)
+
+plot(xVals, yCumVals, type = "l", lwd = 2,
+     main = "CDF for N(2,1)",
+     xlab = "x", ylab = "F(x)")
+
+points(a, 0.05, pch = 19, col = "maroon")
+abline(v = a, col = "maroon", lty = 2)
+abline(h = 0.05, col = "maroon", lty = 2)
+
+dev.off()
+
+
+# 4c
+png("/Users/chris/Maths/MA-223/Oblig/2b/oppg4c.png", width = 1400, height = 700)
+par(mfrow = c(1, 2))
+
+lambda = 4.4
+
+mu_T = 1 / lambda
+sigma_T = 1 / lambda
+P_interval = pexp(0.28, rate = lambda) - pexp(0.15, rate = lambda)
+
+mu_T
+sigma_T
+P_interval
+
+# PDF
+xVals = seq(0, 1.2, 0.001)
+yVals = dexp(xVals, rate = lambda)
+
+plot(xVals, yVals, type = "l", lwd = 2,
+     main = "PDF for Exp(4.4)",
+     xlab = "t", ylab = "f(t)")
+
+shade_x = seq(0.15, 0.28, 0.001)
+shade_y = dexp(shade_x, rate = lambda)
+
+polygon(c(0.15, shade_x, 0.28),
+        c(0, shade_y, 0),
+        col = "plum", border = NA)
+
+lines(xVals, yVals, lwd = 2)
+abline(v = 0.15, col = "maroon", lty = 2)
+abline(v = 0.28, col = "maroon", lty = 2)
+
+# CDF
+yCumVals = pexp(xVals, rate = lambda)
+
+plot(xVals, yCumVals, type = "l", lwd = 2,
+     main = "CDF for Exp(4.4)",
+     xlab = "t", ylab = "F(t)")
+
+F_015 = pexp(0.15, rate = lambda)
+F_028 = pexp(0.28, rate = lambda)
+
+points(c(0.15, 0.28), c(F_015, F_028), pch = 19, col = "maroon")
+abline(v = 0.15, col = "maroon", lty = 2)
+abline(v = 0.28, col = "maroon", lty = 2)
+abline(h = F_015, col = "maroon", lty = 3)
+abline(h = F_028, col = "maroon", lty = 3)
+
+dev.off()
+
+
+# 4d
+png("/Users/chris/Maths/MA-223/Oblig/2b/oppg4d.png", width = 1400, height = 700)
+par(mfrow = c(1, 2))
+
+t_verdi = qt(0.1, df = 4)
+t_verdi
+
+# PDF
+xVals = seq(-5, 5, 0.01)
+yVals = dt(xVals, df = 4)
+
+plot(xVals, yVals, type = "l", lwd = 2,
+     main = "PDF for t(4)",
+     xlab = "x", ylab = "f(x)")
+
+shade_x = seq(-5, t_verdi, 0.01)
+shade_y = dt(shade_x, df = 4)
+
+polygon(c(-5, shade_x, t_verdi),
+        c(0, shade_y, 0),
+        col = "plum", border = NA)
+
+lines(xVals, yVals, lwd = 2)
+abline(v = t_verdi, col = "maroon", lwd = 2)
+
+# CDF
+yCumVals = pt(xVals, df = 4)
+
+plot(xVals, yCumVals, type = "l", lwd = 2,
+     main = "CDF for t(4)",
+     xlab = "x", ylab = "F(x)")
+
+points(t_verdi, 0.1, pch = 19, col = "maroon")
+abline(v = t_verdi, col = "maroon", lty = 2)
+abline(h = 0.1, col = "maroon", lty = 2)
+
+dev.off()
+
+
+# 4e
+png("/Users/chris/Maths/MA-223/Oblig/2b/oppg4e.png", width = 1400, height = 700)
+par(mfrow = c(1, 2))
+
+a = 3
+b = 2
+
+mu_X = a / (a + b)
+sigma_X = sqrt((a * b) / (((a + b)^2) * (a + b + 1)))
+P_interval = pbeta(0.65, a, b) - pbeta(0.4, a, b)
+
+mu_X
+sigma_X
+P_interval
+
+# PDF
+xVals = seq(0, 1, 0.001)
+yVals = dbeta(xVals, a, b)
+
+plot(xVals, yVals, type = "l", lwd = 2,
+     main = "PDF for Beta(3,2)",
+     xlab = "x", ylab = "f(x)")
+
+shade_x = seq(0.4, 0.65, 0.001)
+shade_y = dbeta(shade_x, a, b)
+
+polygon(c(0.4, shade_x, 0.65),
+        c(0, shade_y, 0),
+        col = "plum", border = NA)
+
+lines(xVals, yVals, lwd = 2)
+abline(v = 0.4, col = "maroon", lty = 2)
+abline(v = 0.65, col = "maroon", lty = 2)
+
+# CDF
+yCumVals = pbeta(xVals, a, b)
+
+plot(xVals, yCumVals, type = "l", lwd = 2,
+     main = "CDF for Beta(3,2)",
+     xlab = "x", ylab = "F(x)")
+
+F_04 = pbeta(0.4, a, b)
+F_065 = pbeta(0.65, a, b)
+
+points(c(0.4, 0.65), c(F_04, F_065), pch = 19, col = "maroon")
+abline(v = 0.4, col = "maroon", lty = 2)
+abline(v = 0.65, col = "maroon", lty = 2)
+abline(h = F_04, col = "maroon", lty = 3)
+abline(h = F_065, col = "maroon", lty = 3)
+
+dev.off()
+
+
+# 4f
+png("/Users/chris/Maths/MA-223/Oblig/2b/oppg4f.png", width = 1400, height = 700)
+par(mfrow = c(1, 2))
+
+k = 0.5
+lambda = 3
+
+mu_T = lambda * gamma(1 + 1/k)
+sigma_T = sqrt(lambda^2 * (gamma(1 + 2/k) - gamma(1 + 1/k)^2))
+P_interval = pweibull(4, shape = k, scale = lambda) - pweibull(2, shape = k, scale = lambda)
+
+mu_T
+sigma_T
+P_interval
+
+# PDF
+xVals = seq(0, 20, 0.01)
+yVals = dweibull(xVals, shape = k, scale = lambda)
+
+plot(xVals, yVals, type = "l", lwd = 2,
+     main = "PDF for Weibull(0.5,3)",
+     xlab = "t", ylab = "f(t)")
+
+shade_x = seq(2, 4, 0.01)
+shade_y = dweibull(shade_x, shape = k, scale = lambda)
+
+polygon(c(2, shade_x, 4),
+        c(0, shade_y, 0),
+        col = "plum", border = NA)
+
+lines(xVals, yVals, lwd = 2)
+abline(v = 2, col = "maroon", lty = 2)
+abline(v = 4, col = "maroon", lty = 2)
+
+# CDF
+yCumVals = pweibull(xVals, shape = k, scale = lambda)
+
+plot(xVals, yCumVals, type = "l", lwd = 2,
+     main = "CDF for Weibull(0.5,3)",
+     xlab = "t", ylab = "F(t)")
+
+F_2 = pweibull(2, shape = k, scale = lambda)
+F_4 = pweibull(4, shape = k, scale = lambda)
+
+points(c(2, 4), c(F_2, F_4), pch = 19, col = "maroon")
+abline(v = 2, col = "maroon", lty = 2)
+abline(v = 4, col = "maroon", lty = 2)
+abline(h = F_2, col = "maroon", lty = 3)
+abline(h = F_4, col = "maroon", lty = 3)
+
+dev.off()
+
+# Reset layout afterwards
+par(mfrow = c(1, 1))
+
+
+
+# ============================================================
+# Oppgave 5 - lagrer alle figurer i valgt mappe
+# ============================================================
+
+setwd("/Users/chris/Maths/MA-223/Oblig/2b")
+set.seed(223)
+
+# Reset plotting layout
+par(mfrow = c(1, 1))
+
+# ------------------------------------------------------------
+# 5a) Standard normalfordeling
+# ------------------------------------------------------------
+png("oppg5a_standard_normal.png", width = 1400, height = 900)
+
+x = seq(-4, 4, length = 1000)
+y = dnorm(x, mean = 0, sd = 1)
+
+plot(x, y, type = "l", lwd = 2,
+     main = "Standard normalfordeling",
+     xlab = "x", ylab = "Tetthet")
+
+dev.off()
+
+# ------------------------------------------------------------
+# 5b) Ett tilfeldig trekk
+# ------------------------------------------------------------
+tilfeldig_trekk = rnorm(1, mean = 0, sd = 1)
+tilfeldig_trekk
+
+# ------------------------------------------------------------
+# 5c) 50 trekk + teoretisk kurve
+# ------------------------------------------------------------
+png("oppg5c_50_trekk.png", width = 1400, height = 900)
+
+data_50 = rnorm(50, mean = 0, sd = 1)
+
+hist(data_50, probability = TRUE, breaks = 10,
+     col = "purple",
+     main = "50 trekk fra N(0,1)",
+     xlab = "Verdi")
+
+xVals = seq(-4, 4, 0.01)
+yVals = dnorm(xVals, mean = 0, sd = 1)
+lines(xVals, yVals, col = "maroon", lwd = 2)
+
+dev.off()
+
+# ------------------------------------------------------------
+# 5d) 500 trekk + teoretisk kurve
+# ------------------------------------------------------------
+png("oppg5d_500_trekk.png", width = 1400, height = 900)
+
+data_500 = rnorm(500, mean = 0, sd = 1)
+
+hist(data_500, probability = TRUE, breaks = 20,
+     col = "purple",
+     main = "500 trekk fra N(0,1)",
+     xlab = "Verdi")
+
+xVals = seq(-4, 4, 0.01)
+yVals = dnorm(xVals, mean = 0, sd = 1)
+lines(xVals, yVals, col = "maroon", lwd = 2)
+
+dev.off()
+
+# ------------------------------------------------------------
+# 5d) 50000 trekk + teoretisk kurve
+# ------------------------------------------------------------
+png("oppg5d_50000_trekk.png", width = 1400, height = 900)
+
+data_50000 = rnorm(50000, mean = 0, sd = 1)
+
+hist(data_50000, probability = TRUE, breaks = 40,
+     col = "purple",
+     main = "50000 trekk fra N(0,1)",
+     xlab = "Verdi")
+
+xVals = seq(-4, 4, 0.01)
+yVals = dnorm(xVals, mean = 0, sd = 1)
+lines(xVals, yVals, col = "maroon", lwd = 2)
+
+dev.off()
+
+# ------------------------------------------------------------
+# 5e) Oppgitt kode med N = 100
+# ------------------------------------------------------------
+png("oppg5e_N100.png", width = 1400, height = 900)
+
+N = 100
+h = (rbinom(N, 100, 0.5) - 50) / 5
+hist(h, probability = TRUE, breaks = seq(-8.05, 8.05, 0.1),
+     col = "purple",
+     main = "Standardisert binomisk fordeling, N = 100",
+     xlab = "h")
+xVals = seq(-4, 4, 0.01)
+yVals = 2 * dnorm(xVals, 0, 1)
+lines(xVals, yVals, col = "maroon", type = "l", lwd = 2)
+
+dev.off()
+
+# ------------------------------------------------------------
+# 5f) N = 1000
+# ------------------------------------------------------------
+png("oppg5f_N1000.png", width = 1400, height = 900)
+
+N = 1000
+h = (rbinom(N, 100, 0.5) - 50) / 5
+hist(h, probability = TRUE, breaks = seq(-8.05, 8.05, 0.1),
+     col = "purple",
+     main = "Standardisert binomisk fordeling, N = 1000",
+     xlab = "h")
+xVals = seq(-4, 4, 0.01)
+yVals = 2 * dnorm(xVals, 0, 1)
+lines(xVals, yVals, col = "maroon", type = "l", lwd = 2)
+
+dev.off()
+
+# ------------------------------------------------------------
+# 5f) N = 10000
+# ------------------------------------------------------------
+png("oppg5f_N10000.png", width = 1400, height = 900)
+
+N = 10000
+h = (rbinom(N, 100, 0.5) - 50) / 5
+hist(h, probability = TRUE, breaks = seq(-8.05, 8.05, 0.1),
+     col = "purple",
+     main = "Standardisert binomisk fordeling, N = 10000",
+     xlab = "h")
+xVals = seq(-4, 4, 0.01)
+yVals = 2 * dnorm(xVals, 0, 1)
+lines(xVals, yVals, col = "maroon", type = "l", lwd = 2)
+
+dev.off()
+
+# ------------------------------------------------------------
+# 5f) N = 100000
+# ------------------------------------------------------------
+png("oppg5f_N100000.png", width = 1400, height = 900)
+
+N = 100000
+h = (rbinom(N, 100, 0.5) - 50) / 5
+hist(h, probability = TRUE, breaks = seq(-8.05, 8.05, 0.1),
+     col = "purple",
+     main = "Standardisert binomisk fordeling, N = 100000",
+     xlab = "h")
+xVals = seq(-4, 4, 0.01)
+yVals = 2 * dnorm(xVals, 0, 1)
+lines(xVals, yVals, col = "maroon", type = "l", lwd = 2)
+
+dev.off()
+
+# Reset plotting layout afterwards
+par(mfrow = c(1, 1))
+
+
+#6a)
+x = seq(0, 1, length = 1000)
+y = dbeta(x, 3, 7)
+
+plot(x, y, type = "l", lwd = 2,
+     main = expression(beta(3,7)),
+     xlab = "x", ylab = "Tetthet")
+
+#b)
+par(mfrow = c(3, 3))
+
+verdier = c(1, 2, 5)
+
+for (a in verdier) {
+  for (b in verdier) {
+    x = seq(0, 1, length = 1000)
+    y = dbeta(x, a, b)
+    
+    plot(x, y, type = "l", lwd = 2,
+         main = paste("Beta(", a, ",", b, ")", sep = ""),
+         xlab = "x", ylab = "Tetthet")
+  }
+}