EconProbabilityDistributions

EconProbabilityDistributions - Introductory Statistics...

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1 Introductory Statistics Stats 210 Probability Distributions
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2 The Normal Distribution The Normal Probability Distribution, or Bell Curve, has only 2 parameters Mean Variance It is denoted by: Read: X is distributed normal with mean and variance ) , ( ~ 2 X X N X σμ X μ 2 X σ 2 2 1 2 1 ) ( = X X i x X e x f πσ
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3 Properties of the Normal Distribution The area under the curve sums to 1 It is symmetric around the mean 50% probability to the right of the mean 50% probability to the left of the mean A normal Distribution with mean 0 and variance 1 is called a Standard Normal Distribution Let’s look at some Normal Distributions
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4 0 .1 .2 .3 .4 -10 -5 0 5 10 Different Variances Z ~ N(0,1) X ~ N(0,2) X ~ N(0,4) Areas under Curves sum to 1
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5 .1 .2 .3 .4 -4 -2 0 2 4 6 Different Means 3 X ~ N(3,1) Z ~ N(0,1)
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6 Calculating Probabilities Remember: The Normal Distribution is a probability distribution function of a continuous random variable The probability hat a normally distributed variable equals a particular value is zero P(X=5)=0 But we can calculate probabilities like P(X<5) P(X>5) P(5<X<10)
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7 Calculating Probabilities Assume you want to calculate the following probability of a normally distributed random variable X Probability that X exceeds 4 This probability will be depend on the mean and the variance of the distribution of X We would have to have a table of probabilities for every possible normal distribution That’s a lot of trees…………. .
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8 0 .1 .2 .3 .4 -10 -5 0 5 10 P(X>4) Z ~ N(0,1) X ~ N(0,2) X ~ N(0,4) X ~ N(-5,2) 4
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9 Importance of the Standard Normal Fortunately we can save the trees Every normally distributed random variable can be transformed into a standard normal variable This way we only need one table of probabilities, the standard normal table
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10 The Standard Normal Transformation
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EconProbabilityDistributions - Introductory Statistics...

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