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Lecture Notes Chapter 4 Section 04

# Lecture Notes Chapter 4 Section 04 - MATH1780...

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MATH 1780: Introduction to Probability Instructor: Jason Snyder

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Contents Continuous Random Variables and Their Probability  Distributions Expected Values of Continuous Random variables The Uniform Distribution The Exponential Distribution The Gamma Distribution The Normal Distribution The Beta Distribution The Weibull Distribution Moment-Generating Functions for Continuous Random  Variables
Suppose an experimenter is measuring the life  expectancy X of a transistor. In this case, X can take on an infinite number of  possibilities. We can not assign a positive probability to each  possible outcome of the experiment because, no  matter how small we might make these  probabilities, they would sum to a number greater  than 1 when accumulated.

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Let’s consider a specific example to try to  understand what is going on here. Suppose we measure the life expectancy of 50  transistors of a certain type. 0.406 0.685 4.778 1.725 8.223 2.343 1.401 1.507 0.294 2.230 0.538 0.234 4.025 3.323 2.920 5.088 1.458 1.064 0.774 0.761 5.587 0.517 3.246 2.330 1.064 2.563 0.511 2.782 6.426 0.836 0.023 0.225 1.514 3.214 3.810 3.334 2.325 0.333 7.514 0.968 3.491 2.921 1.624 0.334 4.490 1.267 1.702 2.634 1.849 0.186
Here is a “histogram” for the data on the previous  slide: An approximate equation for the black line in the  histogram is  2 2 1 ) ( x e x f - =

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What if we want to calculate the probability that a  transistor of this type will last for more than 400  hours? Looking at the chart, we notice that probabilities  are indicated by areas under the curve, i.e., an  integral. Thus  dx e X P x 2 4 2 1 ) 400 ( - =
A random variable X is said to be  continuous  if  there is a function f(x) called the  probability  density function , such that 1) f(x)   0 2) 3)   - = 1 ) ( dx x f = b a dx x f b X a P ) ( ) (

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Lecture Notes Chapter 4 Section 04 - MATH1780...

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