Question 1: Discrete Random Variables (15 points)

1.1(Binomial distribution, 7 points)According to IRS, the chances of your tax return being audited are 2 in 100 if your annual income in less than 1 million dollars.

a)If three tax payers are randomly selected with annual income less than 1 million dollars, what is the probability that none will be audited? (1 point)

b)If three tax payers are randomly selected with annual income less than 1 million dollars, what is the probability that exactly one will be audited? (2 points)

c)If three tax payers are randomly selected with annual income less than 1 million dollars, what is the probability that more than one will be audited? (2 points)

d)If three tax payers are randomly selected with annual income less than 1 million dollars, what is the average number of people being audited? (1 point)

e)If three tax payers are randomly selected with annual income less than 1 million dollars, what is the standard deviation of number of people being audited? (1 point)

1.2 (8 points)The bivariate probability distribution of X and Y is given in Table:

Y

X

2

4

5

0.1

0.2

10

0.1

0.2

15

0.3

0.1

a)Obtain the marginal probability distribution of Y. (1 point)

b)Obtain the conditional probability distribution of Y when X=5. (1 point)

c)Are X and Y independent, explain? (1 point)

d)Calculate the covariance of X and Y. (2 points)

e)Calculate the correlation of X and Y. (2 points)

f)Describe the direction and strength of linear association between X and Y. (1 point)

Question 2: (Continuous Random Variables, 25 points)

The Ministry of Education estimates that the SAT scores of students in United States are normally distributed with mean 500 and standard deviation of 25.

a)Find the probability that randomly selected student achieved SAT score between 450 and 475. (5 points)

b)Find the probability that randomly selected student achieved SAT score between 470 and 520. (5 points)

c)Find the probability that randomly selected student achieved SAT score which is more than 550. (5 points)

d)Find the threshold value of SAT score which 10% of all students are above it (5 points)

e)Suppose that a sample of 60 students is randomly selected from United States population, and the sample is normally distributed, what is the probability that the mean SAT score of the sample of students is above 503? (5 points)

Question 3: (Interval Estimation, 10 points)

195

243

132

133

209

400

142

312

221

289

221

162

134

275

355

293

242

458

378

148

278

222

236

178

202

222

334

208

194

135

363

221

449

265

146

215

113

229

221

243

512

193

134

138

209

207

206

310

293

310

237

135

252

365

371

238

232

271

121

134

203

178

180

148

162

160

86

234

244

266

119

259

108

289

328

331

330

227

162

354

304

141

158

240

82

17

357

187

364

268

368

274

278

190

344

157

219

77

171

280