Lecture Notes Chapter 6.pdf - CHAPTER 6 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Recall that a continuous random variable X is a random variable that

Lecture Notes Chapter 6.pdf - CHAPTER 6 SOME CONTINUOUS...

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CHAPTER 6 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Recall that a continuous random variable X is a random variable that takes all values in an interval (or a set of intervals). The distribution of a continuous random variable is described by a density function f ( x ) . A density curve must satisfy that The total area under the curve, by defini- tion, is equal to 1 or 100%, i.e., Z - f ( x ) dx = 1 . The probability of variable values between a and b is the area from a to b under the curve ( a b ), i.e., the area under the curve for a range of values, Z b a f ( x ) dx , is the pro- portion of all observations for that range. The probability of any event is the area under the density curve and above the values of X that make up the event. E XAMPLE 6.1. What value of r makes the following to be valid density curve? 6.1 Continuous Uniform Distribu- tion Being the simplest continuous distribution, uniform distribution Unif [ A , B ] is often called “rectangular dis- tribution” because the density function forms a rect- angle with base B - A and constant height 1 / ( B - A ) . Continuous Uniform Distribution The density function of the continuous uniform ran- dom variable X on the interval [ A , B ] (or, ( A , B ] , [ A , B ) , ( A , B ) ) is f ( x ; A , B ) = 1 B - A , x [ A , B ] (or, ( A , B ] , [ A , B ) , ( A , B ) ) 0 elsewhere . N OTE . The interval may not always be closed. It can be ( A , B ) , ( A , B ] , or [ A , B ) as well. Mean and Variance of Continuous Uniform r.v. The mean and variance of the uniform distribution are μ = A + B 2 and σ 2 = ( B - A ) 2 12 . E XAMPLE 6.2. Suppose that X follows the continuous uniform distribution Unif [ 2 , 7 ] . (a) Find the PDF of X , f ( x ) . (b) Plot the graph of y = f ( x ) . (c) Find the mean and the standard deviation of X . (d) Determine (i) P ( 3 X < 6 ) (a) (ii) P ( X 5 ) (b) (iii) P ( X = 4 ) . (e) Find the value of c such that P ( 2 < X c ) = 0 . 4.
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30 Chapter 6. Some Continuous Probability Distributions 6.2 Normal Distribution Normal Density Function The density of the normal random variable X with mean μ and variance σ 2 is n ( x ; μ , σ ) = 1 2 πσ e - 1 2 ( x - μ σ ) 2 , - < x < where e = 2 . 71828 ... and π = 3 . 1425926 ... . Normal Density Curve a “ bell shaped ” curve. depends upon two parameters for its particular shape: μ : the mean of the distribution (i.e. loca- tion ). σ : the standard deviation of the distribu- tion (i.e. spread or variation ). E XAMPLE 6.3. Given a family of density curves. Which is which? Properties of a Normal Curve The mode, which is the point on the horizontal axis where the curve is a maximum, occurs at x = μ . The curve is symmetric about a vertical axis through the mean μ . The curve has its points of inflection at x = μ ± σ ; it is concave downward if μ - σ < X < μ + σ and is concave upward otherwise. The normal curve approaches the horizontal axis asymptotically as we proceed in either direction away from the mean.
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