RecitationFourSols - Rensselaer Electrical, Computer, and...

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Rensselaer Electrical, Computer, and Systems Engineering Department ECSE 4500 Probability for Engineering Applications Recitation 4 Solutions Wednesday February 17, 2010 Some common random variables For each of the following random variables, plot the pdf or PMF and show that it integrates to one over all x (or sums to one over all n ). Do not use any symbolic math program or numerical method. Work out the answer analytically. 1. Uniform random variable X U ( a, b ) ,with b>a , f X ( x ) , 1 b a [ u ( x a ) u ( x b )] . Ans: Z + −∞ 1 b a [ u ( x a ) u ( x b )] dx = Z b a 1 b a dx = b a b a =1 . 2. Exponential random variable X , with parameter μ> 0 , f X ( x ) , 1 μ exp( x μ ) u ( x ) = 1 μ e x μ u ( x ) (alternative notation). Ans: Z + −∞ 1 μ e x μ u ( x ) dx = Z + 0 1 μ e x μ dx = e x μ | + 0 = 0+1=1 . 3. Gaussian random variable X N ( μ, σ 2 ) , with parameters μ and σ> 0 , f X ( x ) , 1 2 πσ exp[ 1 2 μ x μ σ 2 ] = 1 2 exp 1 2 μ x μ σ 2 (alternative notation). Please make use here of the known integral Z 0 e x 2 2 dx = r π 2 . Ans: Z + −∞ 1 2 exp[ 1 2 μ x μ σ 2 ] dx = 1 2 π Z + −∞ e y 2 2 dy, with the substitution y
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RecitationFourSols - Rensselaer Electrical, Computer, and...

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