Let x be a gaussian random variable with the same

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Unformatted text preview: 9999915. The Φ and Q functions are available on many programable calculators and on Internet websites.1 Some numerical values of these functions are given in Tables 6.1 and 6.2, in the appendix. Let µ be any number and σ > 0. If X is a standard Gaussian random variable, and Y = σX + µ, then Y is a N (µ, σ 2 ) random variable. Indeed, 1 u2 fX (u) = √ exp − 2 2π , so by the scaling rule (3.3), 1 v−µ fX σ σ (v − µ)2 1 √ exp − 2σ 2 2πσ fY (v ) = = , so fY is indeed the N (µ, σ 2 ) pdf. Graphically, this means that the N (µ, σ 2 ) pdf can be obtained from the standard normal pdf by stretching it horizontally by a factor σ, shrinking it vertically by a factor σ, and sliding it over by µ. Working in the other direction, if Y is a N (µ, σ 2 ) random variable, then the standardized version − of Y , namely X = Y σ µ , is a standard normal random variable. Graphically, this means that the standard normal pdf can be obtained from a N (µ, σ 2 ) pdf by sliding it over by µ (so it becomes c...
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