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If $X$ a normal variate with mean $80$ and standard deviation $10$ compute the following probabilites by standardizing. $P(70$$<$X$)$

1 Answer

  • Standard normal distribution:
  • In a standard normal distribution $\mu=0,\sigma ^2=1$
  • The random variable $X$ can be converted to the standard normal variable $Z$ by the transformation
  • $Z=\large\frac{X-\mu}{\sigma}$
  • The probability density function $Z$ is $\phi(z)=\large\frac{1}{\sqrt{2\pi}}$$e^{-\Large\frac{1}{2}z^2};-\infty < Z < \infty$
  • $Z\sim N(0,1)$
Step 1:
$X\sim N(80,10^2)$
Let $Z=\large\frac{X-\mu}{\sigma}$
The probability density function $Z$ is $\phi(z)=\large\frac{1}{\sqrt{2\pi}}$$e^{-\Large\frac{1}{2}z^2};-\infty < Z < \infty$
Step 2:
$P(70 < X)$
When $X=70$
$P(70 < X)=P(-1 < Z)$
$\qquad\qquad=P(0 < Z < 1)+0.5$
answered Sep 19, 2013 by sreemathi.v

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