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Probability Density Functions 1 Notes Spring

# Probability Density Functions 1 Notes Spring - 2 of 15 1.1...

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Unformatted text preview: 2 of 15 1.1 Introduction R: par (mfrow = c (2 , 2) ) R: curve (xˆ2 ,- 10, 10) R: curve (xˆ3 ,- 10, 10) R: curve ( sin (x) ,- 2 * pi , 2 * pi ) R: curve ( sin ( pi * x)/( pi * x) ,- 5, 5)-10-5 5 10 20406080 x x^2-10-5 5 10-1000 500 x x^3-6-4-2 2 4 6-1.0 0.0 0.5 1.0 x sin(x)-4-2 2 4-0.2 0.2 0.6 1.0 x sin(pi * x)/(pi * x) UNIFORM DISTRIBUTION Uniform distribution f ( x ) . Definition 1.1 Occurs when the probability of a continuous random variable is equal across a range of values. Uniform density: dunif(x, min=0, max=1) Useful for graphing, not useful for directly finding probabilities. R Command In R, all PDF’s have a “d” prefix for density. R: curve ( dunif (x , min = 2 , max = 6) , 0 , 8 , ylim = c (0 , + 0.5) , ylab = ”f (x) ” , main = ”Uniform Density f (x) ”) Anthony Tanbakuchi MAT167 Probability density functions 3 of 15 2 4 6 8 0.0 0.1 0.2 0.3 0.4 0.5 Uniform Density f(x) x f(x) Probability is area! FINDING PROBABILITIES FROM DENSITY FUNCTIONS Finding probabilities Area represents probability! P ( x < a ) = Z a-∞ f ( x ) dx (area to the left of a ) P ( a < x < b ) = Z b a f ( x ) dx (area between a and b ) P ( x > a ) = Z ∞ a f ( x ) dx (area to the right of a ) Anthony Tanbakuchi MAT167 4 of...
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Probability Density Functions 1 Notes Spring - 2 of 15 1.1...

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