Lectures 8&9&10 notes - Chapter 6 PROBABILITY...

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Chapter 6 PROBABILITY CONTINUED: CONTINUOUS RANDOM VARIABLES Reference: Devore 7e Chapter 4. 6.1 Continuous random variables range, pdf , cdf , expectation, variance 6.2 Examples of continuous random variables Uniform, exponential, normal, lognormal, weibull, gamma, chi-squared 6.3 Checking fit of a model to data Probability plots Estimating parameters by method of moments 6.4 Several continuous random variables; Joint, marginal, and conditional probability density functions. Covariance and correlation. 6.1 CONTINUOUS RANDOM VARIABLES Defined through probability density function (pdf) f(x) . The range of a continuous random variable is some subset of ( -∞ , + ), but we can always define f(x) = 0 if x is not in the range. P [ a X b ] = R b a f ( x ) dx f ( x ) 0 , R -∞ f ( x ) dx = 1 , P [ X = a ] = 0 cdf F ( x ) = R x -∞ f ( t ) dt f ( x ) = dF dx μ = E [ X ] = R -∞ xf ( x ) dx E [ g ( X )] = R -∞ g ( x ) f ( x ) dx V ar [ X ] = R -∞ ( x - μ ) 2 f ( x ) dx = E [ X 2 ] - μ 2 E [ aX + b ] = aE [ X ] + b V ar [ aX + b ] = a 2 V ar [ X ] Chebyshev’s Inequality still applies. 1
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6.2 EXAMPLES OF CONTINUOUS RANDOM VARIABLES 1) Uniform Devore 7e: p.133. pdf f ( x ) = 1 b - a a x b 0 elsewhere E [ X ] = a + b 2 V ar [ X ] = ( b - a ) 2 12 MINITAB menu: Calc Probability Distributions Uniform... 2) Exponential Devore 7e: p.157. f ( x ) = λ e - λx x 0 0 x < 0 E [ X ] = 1 λ V ar [ X ] = 1 λ 2 Here parameter λ > 0 MINITAB menu: Calc Probability Distributions Exponential... 3) Normal (or Gaussian) N ( μ, σ 2 ) “The Bell Curve” Devore 7e: Sec.4.3. pdf f ( x ) = 1 σ 2 π exp n - 1 2 σ 2 ( x - μ ) 2 o for parameters σ > 0 and μ E [ X ] = μ V ar [ X ] = σ 2 68%, 95%, 99.7% rule (Devore 7e p.151) MINITAB menu: Calc Probability Distributions Normal...
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