Question
An exponential probability distribution has mean equal to minutes per cuslomer: Calculate the following probabilities for ihe distnbution a) Px < 2) b) P(x < 10) c) Plx < 5) d) P(x <8)a) Ptx < 2)-| J (Round lo four decimal places as needed )b) P(x < 10)-D (Round Io four decimal places a5 needed )c) P(x < 51=L J (Round lo Iour decimal placos a5 needed )d) P(x <8)-L (Round t0 Iour decimal places 35 ngeded )
An exponential probability distribution has mean equal to minutes per cuslomer: Calculate the following probabilities for ihe distnbution a) Px < 2) b) P(x < 10) c) Plx < 5) d) P(x <8) a) Ptx < 2)-| J (Round lo four decimal places as needed ) b) P(x < 10)-D (Round Io four decimal places a5 needed ) c) P(x < 51=L J (Round lo Iour decimal placos a5 needed ) d) P(x <8)-L (Round t0 Iour decimal places 35 ngeded )
Answers
Exponential Probability Between 5: 00 PM and 6: 00 PM, cars arrive at Jiffy Lube at the rate of 9 cars per hour $(0.15$ car per minute). The following formula from statistics can be used to determine the probability that a car will arrive within $t$ minutes of 5: 00 PM. $$ F(t)=1-e^{-0.15 t} $$ (a) Determine how many minutes are needed for the probability to reach $50 \%$. (b) Determine how many minutes are needed for the probability to reach $80 \%$.
In problem. 31. We have random variable T. That represents the waiting time for service in a story. This time is exponentially distributed with a mean of five minutes mean equals five minutes for birdie. It's required to find the probability density function ft for the exponential distribution, F f t equals a multiplied by it was about uh minus 80. Where is he is between zero and infant and a equals one. Divided by mu because new equals wonderful. By then here equals one, divided by five. Then the probability density function equals 1/5. To blow it by E. There's about of minus D divided by five. For barbie. We want to find the probability that T is within one standard deviation of the mean. This probability it's known to be equal 68% to prove it and calculated from principles. That's a good. The standard division. First standard deviation equals one, divided by eight. This is for the exponential distribution and we have eight equals 1/5. Then sigma equals five minutes, calculate this probability which means we want to calculate the probability for tea to be greater than um you minor sigma and smaller than you plastic which means we want to calculate the probability 40 to be between zero and 10 minutes. We can use the integral fall from 0 to 10 for fft Fft is given here one divided by five. That by it was about 50 divided by five. The equals this integral gives the same. See to the bottom line is d divided by five, divided by the differentiation of the power minus one divided by five and we have one divided by five from 0 to 10 equals we can cancel this with this, we have minus and then substitute by then first it's it was about minus two minus E. To the power of zero it equals it is about zero, gives one deployed by minus minus, gives one minus E. Is about of minus two. It is a lot of minus two is all one, 4.135 then equals oh point it 65 and it equals 86.5%. Oh it's not 68. Then this is wrong because this is known for the normal distribution, not for the exponential distribution, and the right answer is 68 86 86.5%. This is a common mistake and this is not the answer. This is the final answer. Let's circuit with a green better. Is that an answer of barbie? And this is the final answer of birthday?
It is problem. The probability density function is fx occurred to .5 E. to the power of negative .5 x Axis Square and zero. So first we're calculator to function the big fx. The integration of X from 02 X 3.5 to 2000 negative 20000.5 X Jax. Mhm. And the answer is yeah, Are negative each about negative .5 x. So for a problem a we're going to calculate the probably that Zeros more than X more than .5. And this is the integration from 0.5 FX T X Which is Big F .5 -F steel. And the answer is .2212. The problem be going to calculate the X greer in two As the equation from 2 to infinitive and the answer would be f infinitive minus F two which is used to power the negative one. For problem. See we have this property here is more than X more than green X. Is Seiko tube be to power. So this is Green X. And the Fx times dX, which is one minus E to the negative 10.5 X. Going to calculate when it's going to be 05. So we were just soft. This equation, which is in the heart Uh simplify it so it's each apartment 3.5 X. There's .95 So negative .5 access lock 0.95 so X. After sucking this equation, we got, the value is point 10-6. So this will be our answer.
The person distribution defines that the probability off ex occurrences interval is P off X is equal to mute the x times e to the negative meal over x factorial Here, um, you is eight. Part A X is five. He or five is eight to the five times eat negative aid over five factorial. This is 32,768 times E to the negative eight over 120. This is 0.916 It's larger than 0.5 so it's not a Euro event. Part B. We need to figure out the probabilities when X is 0123 and four. He, off zero, is a 20 Tom's E to the negative eight over zero factorial. This is one. So he to the negative 8/1. This is E to the negative eight. That 0.3 He off one is eight times E to inactive, eight over one factorial. This is eight times either the negative eight. That's 0.0 27. We just use the same formula and we replace X with, uh 23 and four. We got P up to is 0.107 p off three is 0.286 and then last appeal four is 0.573 We want P off at least the fibers, so that's P off X larger than or equal to five. That's one minus p off zero p off one you have to you have three and peel four So one minus 10.3 minus 0.0 27 0.107 0.286 and then minus 0.573 This is 0.9704 It's not on your event. Part C is similar to part B, but Pierre five is not included, so we need to subtract appeal five. From what we got in part B, the p of X larger than five is 0.9004 minus 0.916 This is 0.80 88
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.