Standardized Random Variables

Consider a random variable \(X\) with mean \(\mu\) and standard deviation \(\sigma\). Define a new random variable \(Y\) by
\[Y=\frac{X-\mu}{\sigma}\,.\]

  1. Show that \(Y\) has mean 0 and variance 1.
  2. Show that if \(a \) is some number  \[\mathbf{P}( Y > a) = \mathbf{P}( X > \mu + a\sigma )\]

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