Exam Keys for Probability and Statistics

Well, my current average in Probability and Statistics is somewhere around 35. I don't complain much; the tests were a little rough, but by and large I have simply not been willing to put forth the time that the teacher desires and so I accept the grade that I have. I think I know a little bit more than 35% of the material in the class, but probably not enough to pass. Regardless, my teacher has said that out final grade will be no less that one letter grade below what we get on the final, which is comprehensive. I find this daunting, since we have covered a lot of information. He also says however that 70% of the information on the final wil be taken from old exams, so I if I learn well what is on the old tests then I can at least get a D. I'll still have to retake it (as I should,) but my overall qpa will not suffer as much. So, I am going to work out the old tests here. (I figure if it is electronic I won't lose it by the time I take it again in two semesters.)

Exam Number One

1. 1. How many ways can six people be lined up to get on a bus?

      $${}_{r}P_n=\frac{n!}{\left(n-r\right)!}$$ $${}_{6}P_6=\frac{6!}{\left(6-6\right)!}=6!=720$$

   2. If three persons insist on following each other how many ways are possible?

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   3. If two certain persons refuse to follow each other, how many ways are possible?

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2. How many bridge hands are possible containing four spades, six diamonds, one club and two hearts?

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3. In a certain federal prision it is known that:

   • 2/3 of the inmates are under 25 years of age
   • 3/5 of the inmates are male
   • 5/8 of the inmates are female or 25 years of age or older

   What is the probability that a prisoner selected at random from this prison is female and at least 25 years of age?

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4. From a box containing four black balls and two green balls, three balls are drawn in succession, each ball being replaced in the box before the next draw is made. Find the proability distribution function for the number of green balls.

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5. The total number of hours, measured in units of 100 hours, that a family runs a vaccum cleaner over a period of one year is a continuous random variable x that has a density function:

   f x = { x ; 0 < x < 1 2 - x ; 1 < x < 2 0 ; elsewhere

   Find the probability that over a period of one year a family runs their vaccum cleaner:

   1. Less than 120 hours:

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   2. Between 50 and 100 hours:

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6. An insurance company offers its policy holders a number of different premium payment options. For a selected policy holder, let x be the number of months between successive payments. The cumulative distribution function of x is:

   $$F\left(x\right)=\{\begin{array}{cc}0& ;x<1\\ 0.4& ;1\le x<3\\ 0.6& ;3\le x<5\\ 0.8& ;5\le x<7\\ 1& ;7\le x\end{array}$$

   1. What is the probability mass function?

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   2. Compute P 4 < X < 7 :

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7. Consider the random variables x and y with the joint probability density function given by:

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