**Question 1**

A company, Berkshire Holdings (BH), makes 50 “handmade” tote bags in a month in New York, USA. Each tote bag has a 2% probability of being defective. Let *X* represent the number of defective tote bags that BH produces in a month.

- Explain what a discrete random variable is and how it differs from a continuous random variable. Based on your explanation, is
*X*a discrete random variable? - Of the total number of tote bags produced in a month, what is the probability that more than 3 are defective?
- Your buddy, Wladimir, claims that if two random variables are independent, they must be mutually exclusive. Appraise this statement, and in particular, explain if this statement is true.

**Question 2**

- Explain the concept of unbiasedness and consistency and discuss their differences.
- Let 𝜌^𝑛 be a biased estimator of 𝜌 that is based on the sample size 𝑛. Suppose the variance of 𝜌^ is such that 𝑉ar(𝜌^) → 0. Is 𝜌^ necessarily inconsistent?
- Your friend, Vitali, claims that the estimate 𝑆𝑛

𝑆𝑛 = 1/n ∑(i=1, n) (Xi – Xn¯) (Yi – Yn¯) will converge in probability limit to E [1/n ∑(i=1,n) (Xi-μx) (Yi-μy)]= Cov (X,Y).

Assuming that the random variables 𝑋 and 𝑌 in part (c) are i.i.d., evaluate if his

a claim is true.

**Question 3**

Inspired by ANL321, you have decided to start up a company providing advisory services related to the property market. Your company, www.smartpropertyanalytics.com.sg, will provide your clients with the fair value of properties that are currently on the market. The fair value will be determined by your company’s proprietary statistical approach, which you plan to develop after completing ANL321.

The possible determinants of property prices that you consider for your pricing model are size (in square feet), height (in storey), age (in years), freehold versus leasehold, distance to the nearest MRT station, whether the property is within 1 kilometer to top primary schools or not, and whether the property is within the central area or not.

(*Note*: “Freehold” refers to a perpetual lease on a property. “Leasehold” refers to a property lease that expires after a certain number of years.)

- Implement an appropriate statistical method to uncover the “fair price” of the property.
- Describe how you would evaluate whether your statistical model “fits” the data well.
- Using your statistical approach, how would you determine which property is the most undervalued?
- Using not more than 500 words, write an executive report for your client (a non-expert in statistics) to explain how your method works, what are the potential issues with your approach, and how your client should interpret the results of your fair value estimates.