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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 years) that a seedling tree survives is a random variable $t$ with probability density function defined by$$f(t)=0.05 e^{-0.05 t} \quad \text { for } t \text { in }[0, \infty)$$

$\begin{array}{lll}{\text { (a) } 20} & {\text { (b) } 20} & {\text { (c) } 0.2325}\end{array}$

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, were given a densely function f of tea which is equal to 0.5 e to the negative 0.5 tea. And so we can notice that this follows exactly the format of an exponential distribution that gives us all of these formulas down here to work with. So if we want to find the mean for tea, that's just going to be expectation of tea, which we see is one over Lambda. So we're gonna get one over 10.5, which would give us 20. Next, we're gonna be looking for the standard deviation, which is the same as the square root of the variance. So we're again gonna get one over Lambda and again get 20 for the final part. We want to know the probability that, ah t is between its mean and one standard deviation above the mean. So this is between 20 and 40 so we would write this as cap f of 40 minus cap f of 20. So all we have to do is go ahead and plug 40 and 20 into our half of tea so we would get one minus you to the negative Ah, 0.5 times 40 which would be to. And then we're gonna subtract off one minus e to the negative one. And if you plug this into a calculator, you will get that. The answer is 0.232 five. So this is our final answer.

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