Exercise 4 (Indicator variables). Let (Ω, P) be a probabilityspace and let E ⊆ Ω be an event. The indicator variable of theevent E, which is denoted by 1E , is the RV such that 1E (ω) = 1 ifω ∈ E and 1E(ω)=0ifω∈Ec.Showthat1EisaBernoullivariablewithsuccessprobabilityp=P(E).

Exercise 5 (Variance as minimum MSE). Let X be a RV. Let xˆ ∈ Rbe a number, which we consider as a ‘guess’ (or ‘estimator’ inStatistics) of X . Let E[(X − xˆ)2] be the mean squared error (MSE)of this estimation.

(i) Showthat

E[(X −xˆ)2]=E[X2]−2xˆE[X]+xˆ2 (2) =(xˆ−E[X])2 +E[X2]−E[X]2 (3)=(xˆ−E[X])2 +Var(X). (4)

(ii)ConcludethattheMSEisminimizedwhenxˆ=E[X]andtheglobalminimumisVar(X).Inthissense, E[X ] is the ‘best guess’ for X and Var(X ) is thecorresponding MSE.

Exercise 6. Suppose we have the following sample of Google’sstock price for the past 50 weeks (unit in $ per stock).

320 326 325 318 322 320 329 317 316 331 320 320 317 329 316 308321 319 322 335 318 313 327 314 329 323 327 323 324 314 308 305 328330 322 310 324 314 312 318 313 320 324 311 317 325 328 319 310324

(i) Compute the sample mean x Ì„ and sample standard deviationx.
(ii) Draw the ordered stem-and-leaf display. How many sample valuesare between x ̄ ± s, and

x ̄±2s?
(iii) Give the five-number summary of the sample. Draw thecorresponding box plot.