STAT 3021 Chapter6.pdf - STAT 3021 Spring 2018 Chapter 6...

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STAT 3021 Spring 2018 Chapter 6. Some Continuous Probability Distributions Recall: If a sample space contains an infinite number of possibilities equal to the number of points on a line segment, it is called a continuous sample space. A random variable is called a continuous random variable if it can take on values on a continuous scale. Examples of continuous random variable are: The amount of rain, in inches, that falls in a randomly selected storm. The height of a randomly selected person. The length of time to play 18 holes of golf. Recall: Definition 3.3 6.1 Continuous Uniform Distribution Example 1. (Exercise 3.17 on page 92) A continuous variable X that can assume values between x = 1 and x = b has a density function given by f ( x ) = 1 / 2. (a) Find b . (b) Find P ( X < 1 . 5). (c) Find P ( X 2 | X > 1 . 5). (d) Find x such that P ( X x ) = 0 . 75. 1
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STAT 3021 Spring 2018 Definition: Uniform Distribution Theorem 6.1 Mean and Variance of the uniform distribution Example 2. Let X unif(-3, 5). Find (a) E ( X ) (b) E [( X - 1) 2 ] 6.2 Normal Distribution The distributions of many quantitative random variables are well-approximated by the normal distribution. Normal distributions are specified by μ and σ . Once μ and σ are specified, the normal curve is complete determined. μ : measures the center of the normal distribution. σ : measures the spread of the normal distribution, “the distance between a typical random variable from the center”. 2
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STAT 3021 Spring 2018 Notation: X N ( μ, σ ) means X is normally distributed with mean μ and standard deviation σ . Figure 6.3: two curves are identical in shape but are centered at different positions along the horizontal axis. Figure 6.4: two normal curves with the same mean but different standard deviations. Features of Normal distribution: 3
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STAT 3021 Spring 2018 Figure 6.5: two normal curves having different means and different standard deviations. Clearly, they are centered at different positions on the horizontal axis and their shapes reflect the two different values of σ . Probability density function of normal distribution For now, we are going to assume that μ and σ are known. Later in the course we will learn how to estimate the unknown values of μ and σ . 4
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STAT 3021 Spring 2018 6.3 & 6.4 Areas under the Normal Curve & Applications of the Normal Distribution In Figures 6.3, 6.4 and 6.5, we saw how the normal curve is dependent on the mean and the standard deviation of the distribution under investigation. The area under the curve between any two ordinates must then also depend on the values μ and σ . This is evident in Figure 6.7 below, where we have different shaded regions corresponding to P ( x 1 < X < x 2 ) for two curves with different means and standard deviations.
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