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A technician services mailing machines at companies in the Phoenix area: Depending on the type of malfunction, the service call take 1,2,3 or 4 hours: The different...

Question

A technician services mailing machines at companies in the Phoenix area: Depending on the type of malfunction, the service call take 1,2,3 or 4 hours: The different types of malfunctions occur at about the same frequency:Let x represent the number of hours the service call may take, the probability function f(3) is

A technician services mailing machines at companies in the Phoenix area: Depending on the type of malfunction, the service call take 1,2,3 or 4 hours: The different types of malfunctions occur at about the same frequency: Let x represent the number of hours the service call may take, the probability function f(3) is



Answers

A technician services mailing machines at companies in the Phoenix area. Depending on the type of malfunction, the service call can take one, two, three, or four hours. The different types of malfunctions occur at about the same frequency.
a. Develop a probability distribution for the duration of a service call.
b. Draw a graph of the probability distribution.
c. Show that your probability distribution satisfies the conditions required for a discrete probability function.
d. What is the probability that a service call will take three hours?
e. $A$ service call has just come in, but the type of malfunction is unknown. It is $3 : 00$ p.M. and service technicians usually get off work at 500 p.M. What is the probability that the service technician will have to work overtime to fix the machine today?

In this question, we have to find the expected value of X. What we have with us is a table. First of all, what is the formula for expected value? The formula for expected value of X is nothing but submission off X p o fix submission off XP FX. We have different values effects. If we look at the table, we have number of repair calls as eggs. It will be 012 and three and then we have respective probabilities off 30.1 point 3.4 and point to. Okay, So what is going to be my expected value summation off. So let's just write it. This is going to be zero into 0.1 right, Plus one into 0.3 plus two in 2.43 pointers to in 2.4 plus three in 20.2. What does this become? This turns out to be 1.7, right. So this is 1.7 calls for our 1.7 calls for our alright. Moving on. How do you find obedience? The variance off X is going to be expected. Value off X minus meal square. Right. If I'm not wrong. This is the formula expected value off X minus mu square or one very handy formula is also this is equal toe expected value off X square minus mu square. We can use either of these. So let's try to do with one thing. Let's try to use the first one. What will this become? This will become zero minus 1.7 square in 20.1 plus one minus 1.7 square in 20.3 Full US. Tu minus 1.7 square into 0.4 plus three minus 1.7 square in 20.2. Right. This is going to be the variance. Let me call it V. So if I solve this, the various that I'm getting is 0.67167 So let me just write. This is 0.67 Okay. What is going to be standard deviation? It is going to be Sigma is going to be route off variance. Right. This is going to be somewhere around. If I take my calculator. This is square root off 0.67 which is 0.8185 0.8 185 calls 0.8185 calls per hour for our These are my answers

Okay, so we're told that the time that it takes an I T department fixed the mobile tower is exponentially distributed, So that has an exponential probability distribution. It looks like this in the mind to say X, but we're not told a We're told that the expected time is three hours. Okay, But we need for the exponential and distribution That news also equal told one over, eh? So in that case, we could just solve for a is one third. It actually affects is it's part a f of X is one third, one third kneed in the minus one third X and in part B says, find the probability that it takes between two four hours Rex the time in hours, I guess I could've used tea. What's the matter? So integrate two for this distribution function. And when we get one third defined by negative one third soaking negative needs of the one third ex from before. So this is going to be he too The negative two thirds minus e to the negative four thirds

Really going to solve a problem about 11 here F or fix It was won by name. Eggs Quit zero. There's no negative. That's a new record straight. So a parties f off Mexico's being by BX off a full fix. Pity comes to be f of X equals by nine x. So expectation off executes 0 to 3 except affects, which comes to be by nine x square The X, which is equals toe by 27 Excuse 0 to 3 do thank you by 27 minus boo zero killed by 27 which is then, radiance. The variance off X equals 0 to 3 X squared ffx DX minus expectation. I'll fix 0 to 3 burn in. Excuse me, the X minus two squared. It comes to be too bad. 36 access to for 0 to 3 minus four, which is to three rest for by that is X minus four, which is equal to nine to minus a by foot. Just one by two. Parents off X. Thank you

Okay, so here we have the data given the table and for part A. We want to find the probability that excess Lester equal to three because we want to see what is the problem that at most three lines are in use. So that's given by probably ex less very three, which could be written us. Probably. X equals zero all the way to three, and that can be summed using the values in the table, which is 0.7 now for Part B. We want to figure out what's the probability that there are fewer than three lines in use, So that's X less than three, which is everything except this term here. So that comes out to point for five. And for part C. At least three lines are in use, so X rated the number three, which can be written in a different way. We're just one minus. So we're finding the complimentary probability for ex less than three. So that's one minus 10.45 which is 0.55 and for party between two and five lines. Inclusive are in use. So what does that mean? That means that we're including both influence. Both two and five. So we have necessary to unless you want to hear. So this is one minus probability of X equals one and X equal six and also X equals zero since this one minus 10.15 at this 0.4 minus 0.1, which is 0.7 one. And for party. We want to figure out the probably that's two and four lines into soup are not in use. Okay, so what does that mean? Between two and four lines are not in use. That means we can write it in this, for you can see that, too, is less very cool, too. The complimentary quantity to X, which is six minus x. So it's a writing to lesser equal to X less than or equal to four We're writing two months speaking to six minus excess were before because X is the number of lines in use. So this is the number of lines not in use, so this equal to probably of negative for that's horrible to minus X mystery. Negative, too, Or, in other words, by multiplying that equality by negative one all across, we have two less ripple to X messenger equal to four. So this is what we have here and those. So this is equal to poverty of X equals two plus X equals three plus X equals four, which comes out to 0.65 We go back to the table here, 0.2 point 25.2. So you add them. Get 0.64 wolf a part F. We want to figure out four lines or at these four lines are not in use. So at least four lines not in use. We can use the same kind of logic here. We can say that probably of six minus X. It's greater than or equal to four, which is the same thing is probably be off ex less than or equal to two, which comes out to point one. Let's 0.15 was 0.2, and that sums to 0.45 So that finishes the problem.


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