Question

Let the continuous random variable X denote the current measured in a thin 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) = 0.04 for 0 ≤ x ≤ 20. What is the probability that a current measurement is less than 10 milliamperes?

Answer
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Answer 1

The probability that the current measurement is less than 10 milliamperes is calculated by integrating the probability density function from 0 to 10, yielding a result of 0.4, or 40%.

Step-by-step explanation:

The probability that a current measurement is less than 10 milliamperes can be found by integrating the probability density function (pdf) of the continuous random variable X over the interval from 0 to 10. Given the pdf f(x) = 0.04 for 0 ≤ x ≤ 20, the integral of f(x) from 0 to 10 yields the probability P(X < 10). The calculation is as follows:

∫010 0.04 dx = 0.04x |010 = (0.04)(10) - (0.04)(0) = 0.4

Therefore, the probability that the current measurement is less than 10 milliamperes is 0.4 or 40%.

To find the probability that a current measurement is less than 10 milliamperes, we need to calculate the cumulative distribution function (CDF) for the given probability density function (PDF).

The CDF is defined as the integral of the PDF from negative infinity to the given value. In this case, we want to find P(X<10), which is the integral of f(x) from 0 to 10.

Using the given PDF, the CDF is calculated as:

F(x) = ∫[0, x] f(t) dt = ∫[0, x] 0.04 dt = 0.04x

Therefore, P(X<10) = F(10) = 0.04(10) = 0.4 or 40%.