A research group of engineers has determined that the probability density curve for the proportion of resistors that survive the first test is f(x) = x5 + 6x + 16k for 0 < x < 4 but they need to know the value of k that makes this a valid probability density. What is the value of k?
Added by Patricia B.
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Step 1: Require the total probability to be 1: ∫ from 0 to 4 of (x^5 + 6x + 16k) dx = 1. Show more…
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Resistors labeled 100Ω have true resistances that are between 80Ω and 120Ω Let X be the mass of a randomly chosen resistor. The probability density function of X is given by f(x) = { (x - 80) / 800 80 < x < 120; 0 otherwise } a. What proportion of resistors have resistances less than 90 Ω? Let X be random variable denoting the mass of randomly chosen resistor. The pdf of X is given by f(x) = (x - 80) / 800; 80 < x < 120 a) The proportion of resistors that have resistance less than 90?, P(X < 90) P(X < 90) = ∫[80 to 90] f(x)dx = ∫[80 to 90] (x - 80) / 800 dx = [ 1/800 * (x^2 / 2) - 1/10 * x ] evaluated from 80 to 90 = 850 / 800 - 10 / 10 = 17 / 16 - 1 = 1 / 16 Hence, the proportion of resistors that have resistance less than 90?, P(X < 90) = 1/16
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