Question

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[Z₁ − Z₂ < 2].

c)P[Z1 + Z2 <2].

Answer 1

$a:\\\bar{Z}\sim N\left( 0,\frac{1}{16} \right) \\P\left( \bar{Z}<\frac{1}{2} \right) =P\left( 4\bar{Z}<2 \right) =\varPhi \left( 2 \right) =0.9772\\b:\\Z_1-Z_2\sim N\left( 0,1^2+1^2 \right) =N\left( 0,2 \right) \\P\left( Z_1-Z_2<2 \right) =P\left( \frac{Z_1-Z_2}{\sqrt{2}}<\sqrt{2} \right) =\varPhi \left( \sqrt{2} \right) =0.9214\\c:\\Z_1+Z_2\sim N\left( 0,1^2+1^2 \right) =N\left( 0,2 \right) \sim Z_1-Z_2\\P\left( Z_1+Z_2<2 \right) =P\left( Z_1-Z_2<2 \right) =0.9214$