代做MATH 36A: Probability - Summer 2024帮做Python语言程序

MATH 36A: Probability - Summer 2024

Problems dealing with conditional expectation

1. To solve one of the two problems in the final, students may try 3 different approaches:

• A direct path to the solution, that takes 20 minutes;

• A wrong and brief approach: after 10 minutes, the student realizes it will not work,

• A wrong and painful approach, where the students spends one hour (60 minutes) computing various integrals before realizing it will not work.

We assume that a given student takes any of the approaches uniformly at random (i.e., each approach with the same probability 1/3), and this both at the beginning of the exercise and after a failed attempt (therefore, we make the unlikely assumption that after a failed tentative, they may try again the same approach to make sure it fails, with equal probability).

Compute the expected time for solving the problem.

2. (To get started, you may consider the hint below) A building has N floors. The number of persons in the elevator at the ground floor is a Poisson random variable with parameter λ. Compute the expected number of stops of the elevator.

Hint: start showing that for k people in the elevator, the expected number of stops is N(1 − (1 − 1/N)k). To this end, you may introduce a Bernoulli R.V. Xl = 1 if elevator stops at floor l and 0 otherwise.

3. Random Walk A traveler is lost on a long, 1-dimensional road, with a (possibly unfair) coin in their pocket. Every minute, they toss the coin and decide whether to go one step forward or one step backward.

Mathematically, every minute independently, they go from position x to position x + 1 with probability p (not necessarily equal to 0.5) or backward, from x to x − 1, otherwise.

Prof. Random sees the traveler at position x0 at time t, and returns to fetch the walker after 10 minutes. What is Prof. Chance’s best guess for the position of the traveler?

Limit theorems

Note: Some of the relevant numerical values were pre-computed and provided after the problems.

4. Prof. Chance has lost his kite in a tree at a height of 180 meters. He has 100 students and asks them to climb one on top of the other (each student standing on the head of the one below them) and get the kite for him.

Assuming that the height of students are iid, with a mean of µ = 1.70 meters and a variance σ2 = 0.09 meter2 , compute an (approximate) probability for the students to be able to reach the kite, and explain why you should convince Prof Chance to forget about it!

5. T-mobile estimated that on their 2h plans, all customers exceed time. The excess time used follows an exponential distribution with parameter λ = 1/22 minute −1 .

Consider a sample of 80 customers with this plan.

(a) What is the probability that a randomly chosen customer exceeds time by more than 20 minutes?

(b) What is the probability that, in average among the 80 customers chosen, the average excess time is above 20 minutes?

(c) Explain why these probabilities are different.

Numerical hint: For X a Gaussian N(0, 1), I give P[X > 3/3] ≈ 0.1587, P[X > 10/3] ≈ 4.3 × 10−4 , P[X > 2/√ 80] ≈ 0.4115, P[X > − √ 80/11] ≈ 0.7919