3670Quiz2sol-1 (1).pdf - MATH 3670-N Quiz 2 Name Key Problem 1 Suppose a random variable X has pdf 2 x1 x3 f(x = 0 else a(3 points Find E(X R Rb E(X = 1

# 3670Quiz2sol-1 (1).pdf - MATH 3670-N Quiz 2 Name Key...

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MATH 3670-N Quiz 2 Name: Key Problem 1 Suppose a random variable X has pdf f ( x ) = 2 x 3 , x 1 0 , else. a. (3 points) Find E ( X ) . E ( X ) = R 1 2 x 2 dx = lim b →∞ R b 1 2 x 2 dx = lim b →∞ ( - 2 x ) b 1 = 2 . b. Bonus (1 point) Does the variance of X exist? Explain. E ( X 2 ) = R 1 2 x dx = lim b →∞ 2 ln( b ) = . Hence σ 2 = ∞ - 4 = . So the variance is infinite and hence does not exist as a real number. Problem 2 (4 points) Suppose X is an exponential random variable of mean μ = 2 , that is X has pdf f ( x ) = 1 2 e - x/ 2 , x > 0 0 , else. Find P ( | X - μ | < 2) . P ( | X - 2 | < 2) = P (0 < X < 4) = R 4 0 1 2 e - x/ 2 dx = 1 - 1 e 2 . Problem 3 (4 points) Suppose X is normally distributed with mean 3 and stan- dard deviation 2 . Find P ( - 1 X < 1) . You may use Table 1 on the next page. P ( - 1 X < 1) = P ( - 2 < X - 3 2 < - 1) = P ( - 2 < Z < - 1) = φ ( - 1) - φ ( - 2) = (1 - φ (1)) - (1 - φ (2)) = φ (2) - φ (1) 0 . 97725 - 0 . 84134 = 0 . 13591 .
STANDARD NORMAL DISTRIBUTION: Table Values Represent AREA to the LEFT of the Z score. Z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 0.0 .50000 .50399 .50798 .51197 .51595 .51994 .52392 .52790 .53188 .53586 0.1 .53983 .54380 .54776 .55172 .55567 .55962 .56356 .56749 .57142 .57535 0.2 .57926 .58317 .58706 .59095 .59483 .59871 .60257 .60642 .61026 .61409 0.3 .61791 .62172 .62552 .62930 .63307 .63683 .64058 .64431 .64803 .65173 0.4 .65542 .65910 .66276 .66640 .67003 .67364 .67724 .68082 .68439 .68793 0.5 .69146 .69497 .69847 .70194 .70540 .70884 .71226 .71566 .71904 .72240 0.6 .72575 .72907 .73237 .73565 .73891 .74215 .74537 .74857 .75175 .75490 0.7 .75804 .76115 .76424 .76730 .77035 .77337 .77637 .77935 .78230 .78524 0.8 .78814 .79103 .79389 .79673 .79955 .80234 .80511 .80785 .81057 .81327 0.9 .81594 .81859 .82121 .82381 .82639 .82894 .83147 .83398 .83646 .83891 1.0 .84134 .84375 .84614 .84849 .85083 .85314 .85543 .85769 .85993 .86214 1.1 .86433 .86650 .86864 .87076 .87286 .87493 .87698 .87900 .88100 .88298 1.2 .88493 .88686 .88877 .89065 .89251 .89435 .89617 .89796 .89973 .90147 1.3 .90320 .90490 .90658 .90824 .90988 .91149 .91309 .91466 .91621 .91774 1.4 .91924 .92073 .92220 .92364 .92507 .92647 .92785 .92922 .93056 .93189 1.5 .93319 .93448 .93574 .93699 .93822 .93943

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