MATH
MATH30-6-Lecture-4

MATH30-6-Lecture-4 - Continuous Random Variables and...

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MATH30-6 Probability and Statistics Continuous Random Variables and Probability Distributions

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Objectives At the end of the lesson, the students are expected to Determine probabilities from the probability density functions; Determine probabilities from cumulative distribution functions and cumulative distribution functions from probability density functions, and the reverse; Calculate means and variances for continuous random variables; and Calculate probabilities, determine means and variances for some common continuous probability distributions.
Probability Density Function For a continuous random variable X , a probability density function is a function such that (1) f ( x ) ≥ 0 (2) (3) area under f ( x ) from a to b for any a and b (4-1)

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Probability Density Function A histogram is an approximation to a probability density function. If X is a continuous random variable , for any x 1 and x 2 , P ( x 1 X x 2 ) = P ( x 1 < X x 2 ) = P ( x 1 X < x 2 ) = P ( x 1 < X < x 2 ) (4-2)
Probability Density Function Examples: 4-1/110 Electric Current Let the continuous random variable X denote the current measured in a thin copper wire in milliamperes. Assume that the range of X is [4.9, 5.1] mA, and assume that the probability density function pf X is f ( x ) = 5 for 4.9 ≤ x ≤ 5.1. What is the probability that a current measurement is less than 5 milliamperes?

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Probability Density Function 4-1/111 Suppose that f ( x ) = e x for 0 < x . Determine the following: (a) P (1 < X ) (b) P (1 < X < 2.5) (c) P ( X = 3) (d) P ( X < 4) (e) P (3 ≤ X ) (f) x such that P ( x < X ) = 0.10 (g) x such that P ( X x ) = 0.10.
Probability Density Function 4-2/111 Suppose that f ( x ) = 3(8 x x 2 )/256 for 0 < x < 8. Determine the following: (a) P ( X < 2) (b) P ( X < 9) (c) P (2 < X < 4) (d) P ( X > 6) (e) x such that P ( X < x ) = 0.95. 4-3/111 Suppose that f ( x ) = 0.5 cos x for −π/2 < x < π/2. Determine the following: (a) P ( X < 0) (b) P ( X < −π/4) (c) P (−π/4 < X < π/4) (d) P ( X > −π/4) (e) x such that P ( X < x ) = 0.95.

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Probability Density Function 4-5/111 Suppose that f ( x ) = x /8 for 3 < x < 5. Determine
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• Summer '13
• Math, Statistics, Probability, Probability distribution, Probability theory, probability density function, Cumulative distribution function

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