## Seminar 2. Probabilities and quantiles ## ### Normal distribution # P(Z < 1.96) pnorm(1.96) # P(X < 1.96) X ~ N(2, sigma = 1) pnorm(1.96, mean = 2, sd = 1) # P(-8 < X < 7) X ~ N(6, sigma = 3) pnorm(7, mean = 6, sd = 3) - pnorm(-8, mean = 6, sd = 3) ### Quantiles # quantile of level 0.2 for Z ~ N(0, 1) qnorm(p = 0.2) # quantile of level 0.36 for X ~ N(5, sigma = 2) qnorm(p = 0.36, mean = 5, sd = 2) # Problem 3, part 2 (from seminar09, Moivre-Laplace Theorem) n <- 6 p <- 0.5 q <- 1 - p # P(2 <= S <= 4) pnorm(4, mean = n*p, sd = sqrt(n*p*q)) - pnorm(2, mean = n*p, sd = sqrt(n*p*q)) ### Binomial distribution # P(X <= 6) X ~ Binom(n = 6, p = 0.5) pbinom(q=6, size=6, prob=0.5) # P(X = 6) dbinom(6, size=6, prob=0.5) # Problem 3, part 1 (from seminar09, Exact solution) # P(X = 2) + P(X = 3) + P(X = 4) dbinom(2, size=6, prob=0.5) + dbinom(3, size=6, prob=0.5) + dbinom(4, size=6, prob=0.5) # Binomial coefficient C_n^k # C_10^2, n = 10, k = 2 choose(10, 2) ### Random samples and histograms # random sample of n = 20 from Z ~ N(0, 1) v <- rnorm(20) v # histogram - does not look like normal as n = 20 is very small hist(v) # random sample of n = 1000 from Z ~ N(0, 1) v <- rnorm(1000) # histogram - looks like normal as we increase n to 1000 hist(v) # add color hist(v, col = "hotpink") # add title hist(v, col = "hotpink", main = "Моя первая розовая гистограмма")