Highly Recommended: For both parts below, go back to Homework 2 Question 2 and see how you worked with sample minima there. That approach will be very useful but in a different way for each part.
a) For 1 ≤ i ≤ 5 , let Xᵢ have the geometric (pᵢ) distribution on {1,2,3,...}. Suppose X₁,X₂,...,X₅ are independent. Let M=min{Xᵢ:1 ≤ i ≤ 5} .
• Find the distribution of M. Identify it as one of the famous ones and provide its name and parameters. You are welcome to use the notation qᵢ=1-pᵢ
• Hence find the expectation of M.
b) There are 300 students in a probability class. Each student tosses a coin 100 times and notes the number of heads. Let S be the smallest of the 300 numbers noted.
• Let F be the cdf of the binomial (100, 0.5) distribution. Write a math formula for E(S) in terms of F. Explain your formula.
• Complete the code cell below by writing an expression that evaluates to E(S). To write just one expression, use array operations. It helps that if k is an array, stats.norm.cdf(k, 100, 0.5) evaluates to an array containing the binomial cdf evaluated at each element of k.
