Once the errors are determined we can determine the by combining them through uI 2 2 2 2 u u u u j L K T j If more than one instrument is used uI 1 v uI 2 uT uI21 uI22 v uT Let s take a first shot at instrument uncertainty A pressure transducer is connected to a digital panel meter The panel meter converts the pressure transducer s output in volts back to pressure in psi The manufacturer provides the following information about the panel meter Resolution 0 1 psi Linearity Within 0 1 of reading Drift Less than 0 1psi 6 month within 32 900F For the transducer the data sheet states an accuracy of within 0 5 of it s reading Estimate the combined instrument uncertainty in the measured pressure at a nominal value of 100 psi at room temperature Assume the transducer s response is linear with an output of 1V psi and it is less than 6 months old Resolution 0 1 psi Linearity Within 0 1 of reading Drift Less than 0 1psi 6 month within 32 900F For the transducer the data sheet states an accuracy of within 0 5 of it s reading Estimate the combined instrument uncertainty in the measured pressure at a nominal value of 100 psi at room temperature Assume the transducer s response is linear with an output of 1V psi and it is less than 6 months old What is the uncertainty in the transducer uT 0 005 100 psi 0 50 psi Resolution 0 1 psi Linearity Within 0 1 of reading Drift Less than 0 1psi 6 month within 32 900F Estimate the combined instrument uncertainty in the measured pressure at a nominal value of 100 psi at room temperature Assume the transducer s response is linear with an output of 1V psi and it is less than 6 months old uR 0 1 0 5 0 05 psi Display uL 0 001 100 psi 0 10 psi uD 0 1 psi 6months 6months 0 10 psi uP uR2 uL2 uD2 0 15 psi Now we can simply combine these as uT uP2 uT2 0 52 0 152 0 52 psi Need to look closely at data sheets Outcome of an experiment given by value X If X changes randomly from experiment to experiment it is called a random variable In practice nearly all measured variables are random variables Continuous random variable can take on any points along the real line Current in a copper wire Length of a part Pressure Time Temperature Discrete random variable takes on only discrete specific values Number of bits transmitted in error in a sequence Number of scratches on a surface Proportion of defective parts out of 1000 f x Probability Distribution of X set of probabilities associated with the possible values of X The probability density function f x can be used to describe the probability distribution of the continuous random variable X Length mm Once we have it we can find the probability through b P a x b f x dx a f x 0 The properties of the PDF are f x 1 IMPORTANT POINT f x is used to calculate an area that represents the probability that X assumes a value in a b Let x be the current measured in a copper wire in milliamperes Assume that the range of X is 0 20 mA and assume that the probability density function of X is f x 0 05 0 x 20 A Find P X 10 B Find P 5 X 10 This is pretty easy to see if we sketch out the PDF Another way to describe the probability distribution of a random variable is the Cumulative Distribution Function F x F x P X x x f u du What is P X 1 The CDF can be related to the PDF through b b a a P a X b f x dx f x dx f x dx F b F a The distance in mm from the start of a track on a magnetic disk until the first surface flaw is given by the CDF x F x 1 exp 2000 x 0 A Determine P X 1000 B Determine P 1000 X 2000 P 1000 X 2000 P X 2000 P X 1000 F 2000 F 1000
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