Working with discrete probability distributions

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Alienboy
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18 Mar 2012, 6:18 pm

Here are a few problems I attempted working out on my own, but couldn't understand them enough to get the correct answers:

1. The probabilities that a batch of 4 computers will contain 0,1,2,3 & 4 defective computers are 0.4979, 0.3793, 0.1084, 0.0138 & 0.0007, respectively. Find the standard deviation for the probability distribution. (The answer key reads that the correct answer is 0.73). This is the formula I used and the work I did:

(x-μ)2(squared) x P(x)

0-0.64(squared) x 0.4979 = -.2039
0-0.64(squared) x 0.3793 = -.1553
0-0.64(squared) x 0.0138 = -.0056
0-0.64(squared) x 0.0007 = -.2.867
-3.2762
I know I did this completely wrong because the answer should be 0.73. Can anyone explain this to me?

2. Assume that a procedure yields a binaomial distribution with a trial repeated n times. Use the binomial probability formula to find the probability of x successes given the probability p of success on a single trial. Round to three decimal places.
n = 4, x = 3, p=1/6 I tried using the 0:binompdf( formula option on my TI-83 but I keep getting a syntax error message.

3. Suppose a law enforcement group studying traffic violations determines that the accompanying table describes the probability distribution for five randomely selected people, where x is the number that have received a speeding ticket in the last 2 years. Is it unusual to find no speeders among five randomely selected people?

x P(x)
0 0.08
1 0.18
2 0.25
3 0.22
4 0.19
5 0.08

A. Yes B. No The answer key has B. No as the correct answer, but I am not sure how they got the answer.

4. Use the Poisson Distribution to find the indicated probability. Sunita's job is to provide technical support to computer users. Suppose the arrival of calls can be modeled by a Poisson distribution with a mean of 4.7 calls per hour. What's the probability that in the next 10 minutes there will be 2 or more calls? The answer key has 0.1852 as the correct answer. I keep trying the poissonpdf( option on my TI-83, but I keep getting the wrong answer.

I would appreciate any help from you math wizzes on here. Even a little help would be great. =)


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Alienboy
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18 Mar 2012, 10:04 pm

Nobody can seriously help me out a little? I posted my questions in a few Stats forums, but the people there are cold and ignore me completely.


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OddDuckNash99
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19 Mar 2012, 7:35 am

I can help you with some of the questions. For the first one, you aren't using the correct equation for standard deviation. When you want to find the SD, there always is a square root in the formula. Variance is the SD squared, which is why it doesn't have a square root in the equation. Here's a Wiki link to the classic SD formula: http://en.wikipedia.org/wiki/Standard_deviation

For the third question, the one about the speeding, the answer is "no," because the probability given is a number that is high enough likely to be found in nature. In the data you typed, "x" is how many people they randomly sampled, and "P(x)" is the probability that "x" number of random subjects have indeed received a speeding ticket in the past two years. When "x" is 5, meaning that 5 people have been randomly selected, the probability that ALL of those 5 people received a speeding ticket is very low, only a 0.08% chance. This is the same probability of NOBODY receiving a speeding ticket (when "x" is 0). However, even though 0.08 is a fairly low probability, in statistics, this number is NOT seen as all that low.

When doing hypothesis testing when you run an experiment, the most common alpha-level chosen is 0.05. The alpha-level is the probability that you found significant results when they really weren't any (formally known as a "type-I error"). It's common for statisticians to accept that they have significant results when there is only a 0.05 (or less) chance that they are incorrect. Here, the probability that there are NO speeders in a group of five randomly selected people is 0.08. So, in statistics, that isn't an unusually low number, even though it may sound that way in everyday talk.

As for questions 2 and 4, I'm not familiar with the Poisson distribution, and while I've probably worked with the binomial probability formula, the name doesn't sound familiar. My strength in stats (as with everything in science) is conceptual concepts, NOT the math.


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Alienboy
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19 Mar 2012, 11:31 am

OddDuckNash99 wrote:
I can help you with some of the questions. For the first one, you aren't using the correct equation for standard deviation. When you want to find the SD, there always is a square root in the formula. Variance is the SD squared, which is why it doesn't have a square root in the equation. Here's a Wiki link to the classic SD formula: http://en.wikipedia.org/wiki/Standard_deviation

For the third question, the one about the speeding, the answer is "no," because the probability given is a number that is high enough likely to be found in nature. In the data you typed, "x" is how many people they randomly sampled, and "P(x)" is the probability that "x" number of random subjects have indeed received a speeding ticket in the past two years. When "x" is 5, meaning that 5 people have been randomly selected, the probability that ALL of those 5 people received a speeding ticket is very low, only a 0.08% chance. This is the same probability of NOBODY receiving a speeding ticket (when "x" is 0). However, even though 0.08 is a fairly low probability, in statistics, this number is NOT seen as all that low.

When doing hypothesis testing when you run an experiment, the most common alpha-level chosen is 0.05. The alpha-level is the probability that you found significant results when they really weren't any (formally known as a "type-I error"). It's common for statisticians to accept that they have significant results when there is only a 0.05 (or less) chance that they are incorrect. Here, the probability that there are NO speeders in a group of five randomly selected people is 0.08. So, in statistics, that isn't an unusually low number, even though it may sound that way in everyday talk.

As for questions 2 and 4, I'm not familiar with the Poisson distribution, and while I've probably worked with the binomial probability formula, the name doesn't sound familiar. My strength in stats (as with everything in science) is conceptual concepts, NOT the math.


Yeah I actually wound up figuring out problem 3 on my own. I still appreciate your help setting me on the right path for problem 1. =)


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