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Find (a) the mean of the distribution, (b) the standard deviation of the distribution, and (c) the probability that the random variable is between the mean and 1 standard deviation above the mean. The length of time (in days) required to learn a certain task is a random variable $t$ with probability density function defined by $$f(t)=\frac{e^{-l / 3}}{3} \text { for } t \text { in }[0, \infty)$$

(a) 3$\qquad$ (b) 3$\qquad$ (c) 0.2325

Calculus 1 / AB

Chapter 18

Probability and Calculus

Section 3

Special Probability Density Functions

Continuous Functions

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Okay, So for this question, we're gonna notice that our density function here you to the negative t over three, divided by three is exactly the form of an exponential distribution with Lambda equals 1/3. So if we want to because this is an exponential distribution. We have these formulas down here that we can make use off. So if we want to find the expectation of T, which is the mean we know, it's just one over Lambda. So that will be three If we want to find the standard deviation. We know that's just the square root of the variants which will be one over lambda again. So three again. And if we want to find the probability that, uh, rt is within one standard deviation of its mean, uh from above, that's going to be the probability that T is between three and three plus three. So between three and six and that follows exactly this formula down here. So we know that that's going to be 0.23 25 So these are our final answers

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