Assignment 3

Question 1 (Exercise 6.29)

A distributor of bean seeds determines from extensive tests that 1% of a large batch of seeds will not germinate. She sells the seeds in packets of 200 and guarantees that at least 98% of the seeds will germinate.

  1. Find the probability that any particular packet violates the guarantee

    1. exactly, using a binomial distribution;

    2. using the Poisson approximation.

  2. A gardener buys 13 packets. What is the probability that at least one packet violates the guarantee? In your calculation, you should use the exact probability from the previous part, rather than the approximation.

Question 2 (Exercise 6.7)

Use the equality \[\exp(\lambda)=\sum_{x=0}^{\infty} \frac{\lambda^x}{x!}\] to show that \(\sum_{x=0}^{\infty} p(x) = 1\) when \(X \sim \mathrm{Po}(\lambda)\).

Question 3 (comparative judgement question)

Write a summary, of around forty words, explaining your current best understanding of what a random variable is. Feel free to use any exampples or related functions to help you illustrate your explanation. The best summaries are ones that use key ideas, rather than repeating the original definition.

Next week, you will be asked to look at pairs of these summaries, submitted by your peers, and select the summary which you think best encompasses this idea.