Answer to Question #319530 in Statistics and Probability for mady

Question #319530

Consider independent random variables Zi ∼ N(0, 1), i = 1, …, 16, and let Z be the sample mean. Find:

a) P[Zbar <1/ 2].

b) P[Z1Z2 < 2].

c)P[Z1 + Z2 <2].



1
Expert's answer
2022-04-01T04:36:53-0400

a:ZˉN(0,116)P(Zˉ<12)=P(4Zˉ<2)=Φ(2)=0.9772b:Z1Z2N(0,12+12)=N(0,2)P(Z1Z2<2)=P(Z1Z22<2)=Φ(2)=0.9214c:Z1+Z2N(0,12+12)=N(0,2)Z1Z2P(Z1+Z2<2)=P(Z1Z2<2)=0.9214a:\\\bar{Z}\sim N\left( 0,\frac{1}{16} \right) \\P\left( \bar{Z}<\frac{1}{2} \right) =P\left( 4\bar{Z}<2 \right) =\varPhi \left( 2 \right) =0.9772\\b:\\Z_1-Z_2\sim N\left( 0,1^2+1^2 \right) =N\left( 0,2 \right) \\P\left( Z_1-Z_2<2 \right) =P\left( \frac{Z_1-Z_2}{\sqrt{2}}<\sqrt{2} \right) =\varPhi \left( \sqrt{2} \right) =0.9214\\c:\\Z_1+Z_2\sim N\left( 0,1^2+1^2 \right) =N\left( 0,2 \right) \sim Z_1-Z_2\\P\left( Z_1+Z_2<2 \right) =P\left( Z_1-Z_2<2 \right) =0.9214


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