January_4 - Uniform distribution (I) The density function...

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1 Uniform distribution (I) The density function of a uniform random variable X : 1 ( ) , , 0 elsewhere fx A x B BA = = The mean ( expected value ) of X : () 2 BB AA xdx xf x dx = + = ∫∫ f ( x ) x B A 1
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2 Uniform distribution (II) The cumulative distribution function of X : ( ) ( ) 0 , , 1 Fx PX x x A xA AxB BA Bx = ≤= = = The variance of X : ( ) 2 () 12 VX = F ( x ) x B A 1
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3 Normal distribution (I) The density function of a normal random variable X : ( ) 2 2 2 1 ( ) , , 2 x fx e x µ σ σπ = −∞≤ ≤∞ The variance of X : 2 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 -25 -5 15 35 55 x f(x) The mean of X :
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4 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 -25 -5 15 35 55 x f(x) Normal distribution (II) Same variance, different means!
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5 Normal distribution (III) Same mean, different variances! -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 -25 -5 15 35 55 x f(x)
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6 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 -25 -5 15 35 55 75 95 x f(x) Normal distribution (IV) Different means and different variances!
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7 Normal distribution (V) Standard Normal Distribution ( µ =0, σ 2 =1) Table Look up the Standard Normal Distribution Table, for Φ (z) which is the probability that the standardized normal variable chosen at random is less than a specified value of z . Consider a normal r.v. with mean and variance 2 ( ) 12 Px X x ≤≤ = 21 xx µµ σσ −−  −Φ 
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January_4 - Uniform distribution (I) The density function...

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