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RecitationFour - X ∼ N μ σ 2 with parameters μ and...

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Rensselaer Electrical, Computer, and Systems Engineering Department ECSE 4500 Probability for Engineering Applications Recitation 4 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 )] . 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). 3. Gaussian random variable X
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Unformatted text preview: X ∼ N ( μ, σ 2 ) , with parameters μ and σ > , f X ( x ) , 1 √ 2 πσ exp[ − 1 2 μ x − μ σ ¶ 2 ] = 1 √ 2 πσ exp − 1 2 μ x − μ σ ¶ 2 (alternative notation). You can make use here of the known integral Z ∞ e − x 2 2 dx = r π 2 . 4. Binomial random variable K , with parameters n and p, n a positive integer and ≤ p ≤ 1 . De f ne q , 1 − p. P K ( k ) , ½ ¡ n k ¢ p k q n − k , ≤ k ≤ n, , else. 5. Poisson random variable N, with paramter μ > , P N ( n ) , μ n n ! e − μ u ( n ) . 1...
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