PAge 172, #2

A radioactive particle emits alpha-particles at a rate described by f(t) = 0.1 * exp(-0.1t). Find the probability a particle is emitted within the first 10 seconds given….

  1. No particle is emitted in the first second P(t <= 10 | t < 1) => P(1 <= t <= 10)
func_a <- function(t){
  0.1*exp(-0.1*t)
}

prob_a = integrate(func_a, lower = 1, upper = 10)
prob_a
## 0.536958 with absolute error < 6e-15
  1. No particle is emitted in the five seconds P(t <= 10 | t > 5) => P(5 <= t <= 10) = integral(f(t), a = 5, b = 10)
func_b <- function(t) {
  0.1*exp(-0.1*t)
}

prob_b <- integrate(func_b, lower = 5, upper = 10)
prob_b
## 0.2386512 with absolute error < 2.6e-15
  1. A particle IS emitted witihn the first 3 seconds P( t <= 10 | t <= 3) => integral(f(t), a = 0, b = 3) / integral(f(t), a = 0, b = 10)
func_c_3 <- function(t) {
  0.1*exp(-0.1*t)
}

func_c_10 <- function(t) {
  0.1*exp(-0.1*t)
}

prob_c_i <- integrate(func_c_3, lower = 0, upper = 3) 
prob_c_ii <- integrate(func_c_10, lower = 0, upper = 10)

prob_c_i
## 0.2591818 with absolute error < 2.9e-15
prob_c_ii
## 0.6321206 with absolute error < 7e-15
prob_c <- 0.2591818 / 0.6321206  
prob_c
## [1] 0.4100195
  1. A particle is emitted within 20 seconds P(t <= 10 | t <= 20) => integral(f(t), a = 0, b = 10) / integral(f(t), a = 0, b = 20)
func_d <- function(t) {
  0.1*exp(-0.1*t)
}

prob_d_i <- integrate(func_d, lower = 0, upper = 10)
prob_d_ii <- integrate(func_d, lower = 0, upper = 20)

prob_d_i
## 0.6321206 with absolute error < 7e-15
prob_d_ii
## 0.8646647 with absolute error < 9.6e-15
prob_d <-  0.6321206  / 0.8646647  
prob_d
## [1] 0.7310586