lecture%204_s - APPENDIX B Fundamentals of probability(cont...

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APPENDIX B Fundamentals of probability (cont.)
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Some common continuous random variables
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1. THE NORMAL DISTRIBUTION The Normal distribution describes the probability distribution for a random variable X that has the following properties: X can take any value on the real line (−∞, ∞) The pdf of X has the familiar ‘bell shaped’ curve If ? ? = 𝜇 ?𝑛? 𝑉?𝑟 ? = 𝜎 2 , then the pdf for X can be written as: ? 𝑥 = 1 𝜎√2𝜋 exp 𝑥 − 𝜇 2 2𝜎 2 , −∞ < 𝑥 < ∞ ?~𝑁(𝜇, 𝜎 2 )
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1. THE NORMAL DISTRIBUTION
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2. THE STANDARD NORMAL DISTRIBUTION The Normal distribution has some very nice properties. One of them is that a linear function of a normal random variable is also normal. Hence, if ?~𝑁 𝜇, 𝜎 2 ?𝑛? ? = ? + ??, 𝑡ℎ?𝑛 ?~𝑁(? + ?𝜇, ? 2 𝜎 2 ) So ? = 𝑋−𝜇 𝜎 ~𝑁(0,1) i.e. Z has a standard normal distribution. The pdf of a standard normal random variable Z is given by: 𝜙 𝑧 = 1 √2𝜋 exp 𝑧 2 2 , −∞ < 𝑧 < ∞
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2. WHY IS THE STANDARD NORMAL DISTRIBUTION SO USEFUL?
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