data_605_hw8

Problem 11 (p. 303)

A company buys 100 lightbulbs, each of which has an exponential lifetime of 1000 hours. What is the expected time for the first of these bulbs to burn out? (See Exercise 10.)

mu <- 1000 
bulb <- 100

expected_time = mu / bulb
print(paste0(glue("The first of these bulbs is expected to burn out in about ", {expected_time}, " hours")))

## [1] "The first of these bulbs is expected to burn out in about 10 hours"

Problem 14 (p. 303)

Assume that \(X_1\) and \(X_2\) are independent random variables, each having an exponential density with parameter \(\lambda\). Show that \(Z = X_1 - X_2\) has density \(f_Z(z) = (1/2)\lambda e^{-\lambda |z|}.\)

Read p.291 to 293 and watched https://www.youtube.com/watch?v=f8Nli1AfygM as guide, but I still can’t fully understand the solution.

Problem 1 (p. 320 - 321)

Let \(X\) be a continuous random variable with mean \(\mu = 10\) and variance \(\sigma^2 = 100/3\). Using Chebyshev’s Inequality, find an upper bound for the following probabilities.

mu = 10
sigma_squared = 100 / 3

x = c(2, 5, 9, 20)

eval = map(map(x, function(x) x^2), 
           function(y) sigma_squared / y) %>%
    unlist

eval2 = ifelse(eval >1, 1, eval) %>% round(., 2)

The upper bound for \(P(|X-10|\geq 2)\) is about 1.

The upper bound for \(P(|X-10|\geq 5)\) is about 1.

The upper bound for \(P(|X-10|\geq 9)\) is about 0.41.

The upper bound for \(P(|X-10|\geq 10)\) is about 0.08.