Week 8 probability

library(tidyverse)

## Warning: package 'ggplot2' was built under R version 4.3.2

Question 11 Page 303

# Set the parameters
num_bulbs <- 100
lifetime <- 1000

# Expected time
lifetime / num_bulbs  

## [1] 10

Question 14 Page 303

Given that \(X_1\) and \(X_2\) are independent random variables, each with an exponential density function with parameter \(\lambda\), we aim to find the density function \(f_Z(z)\) for \(Z = X_1 - X_2\).

Case 1: \(z < 0\)

\[ \begin{aligned} f_Z(z) &= \int_{-\infty}^{z} \lambda e^{-\lambda x} \cdot \lambda e^{\lambda (z - x)} \, dx \\ &= \lambda^2 e^{\lambda z} \int_{-\infty}^{z} e^{-2\lambda x} \, dx \\ &= \lambda^2 e^{\lambda z} \left[ \frac{-1}{2\lambda} e^{-2\lambda x} \right]_{-\infty}^{z} \\ &= \frac{1}{2} \lambda e^{\lambda z} \end{aligned} \]

Case 2: \(z \geq 0\)

\[ \begin{aligned} f_Z(z) &= \int_{-\infty}^{z} \lambda e^{-\lambda x} \cdot \lambda e^{\lambda (z - x)} \, dx \\ &= \lambda^2 e^{\lambda z} \int_{-\infty}^{z} e^{-2\lambda x} \, dx \\ &= \lambda^2 e^{\lambda z} \left[ \frac{-1}{2\lambda} e^{-2\lambda x} \right]_{-\infty}^{z} \\ &= \frac{1}{2} \lambda e^{-\lambda z} \end{aligned} \]

Now, combining both results, we get:

\[ f_Z(z) = \begin{cases} \frac{1}{2} \lambda e^{\lambda z} & \text{for } z < 0 \\ \frac{1}{2} \lambda e^{-\lambda z} & \text{for } z \geq 0 \end{cases} \]

\[ f_Z(z) = \frac{1}{2} \lambda e^{-\lambda |z|} \]

This matches the given density function for \(f_Z(z)\), verifying the result for both negative and positive values of \(z\).

Question 1 Page 320

mu <- 10
variance <- 100/3

# k for different probabilities
k_values <- c(2, 5, 9, 20)

# Upper bounds for the probabilities using Chebyshev's Inequality
prob_bounds <- variance / k_values^2

# Output the results
for (i in 1:length(k_values)) {
  cat("Upper bound for P(|X - 10| ≥", k_values[i], "):", prob_bounds[i], "\n")
}

## Upper bound for P(|X - 10| ≥ 2 ): 8.333333 
## Upper bound for P(|X - 10| ≥ 5 ): 1.333333 
## Upper bound for P(|X - 10| ≥ 9 ): 0.4115226 
## Upper bound for P(|X - 10| ≥ 20 ): 0.08333333

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