MY MAIN PROBLEM IS FIGURING OUT WHAT DISCRETE DISTRIBUTION TO USE, BERNOULLI, BINOMIAL, DISCRETE UNIFORM, GEOMETRIC NEGATIVE BINOMIAL, OR POISSON. Every time I choose one, it's the incorrect one. Is there some way I can easily find out which one to use.
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3.40
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Let Y denote a random variable that has a geometric distribution, with a probability of success on any trial denoted by p.
a) Find P(Y>=2) if p=0.1
b) Find P(Y>4 | Y>2) for general p. Compare this result with the unconditional probability P(Y>=2).[This property is referred to as "lack of memory"]
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3.52
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The telephone lines coming into an airline reservation office are all occupied about 60% of the time.
a) if you are calling this office, what is the probability that you will complete your call on the first try?the second try? the third try?
b)If both you and a friend must complete separate calls to this reservation office, what is the probability that it will take a total of four tries for the two of you?
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3.58
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In the article cited in Exercise 3.57, the projected fatality rate for 1975 if the NMSL had not been in effect was 25 per 10^9 vehicle miles. Assume that these conditions had prevailed.
a) Find the probability that at most 15 fatalities occurred in a given block of 10^9 vehicle miles.
b) Find the probability that at least 20 fatalities occurred in a given block of 10^9 vehicle miles.
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3.66
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The number of bacteria colonies of a certain type in samples of polluted water has a Poisson distribution with a mean of two per cubic centimeter.
a)If four 1-cubic-centimeter samples of this water are independently selected, find the probability that at least one sample will contain one or more bacteria colonies.
b)How many 1-cubic-centimeter samples should be selected to establish a probability of approximately 0.95 of containing at least one bacteria colony?
To determine which discrete distribution to use, you need to consider the characteristics of the problem at hand. Here is a brief explanation of each distribution and when they are typically used:
1. Bernoulli Distribution: Use the Bernoulli distribution when there are exactly two possible outcomes, usually denoted as "success" and "failure", with a fixed probability of success (p) for each trial.
2. Binomial Distribution: Use the Binomial distribution when there are a fixed number of independent Bernoulli trials, with each trial having the same probability of success (p). The Binomial distribution represents the number of successful trials in a fixed number of trials.
3. Discrete Uniform Distribution: Use the Discrete Uniform distribution when all outcomes are equally likely. This distribution is commonly used when selecting a random number from a finite set.
4. Geometric Distribution: Use the Geometric distribution when you want to know the number of independent Bernoulli trials required until the first success. It represents the number of failures before the first success.
5. Negative Binomial (Geometric Negative Binomial) Distribution: Use the Negative Binomial distribution when you want to know the number of independent Bernoulli trials required until a fixed number of successes (r) have occurred. It represents the number of failures before r successes.
6. Poisson Distribution: Use the Poisson distribution when you want to model the number of events that occur in a fixed interval of time or space. It is often used when dealing with rare events and has a single parameter lambda (Ξ») which represents the average rate of occurrence.
Now, let's go through each problem and determine which distribution to use:
3.40:
a) This problem asks for the probability P(Y>=2). Since Y is a geometric distribution, we need to use the geometric distribution formula. P(Y>=2) = 1 - P(Y<2). Therefore, we need to use the Geometric distribution.
b) This problem asks for P(Y>4 | Y>2), which is conditional probability given Y>2. Since it involves a conditional probability, we can again use the geometric distribution formula. However, P(Y>=2) is the unconditional probability mentioned in the problem, so we need to compare the result with P(Y>=2) to show the "lack of memory" property. Therefore, we still use the Geometric distribution.
3.52:
a) This problem asks for the probability of completing the call on the first, second, and third tries. Since the probability of completing the call changes after each try, we need to use the geometric distribution.
b) This problem involves the total number of tries for two people. Since it is about the total sum of tries for two people, we need to use the Negative Binomial (Geometric Negative Binomial) distribution.
3.58:
a) This problem asks for the probability of at most 15 fatalities. Since it involves counting the number of occurrences within a given block, we need to use the Poisson distribution.
b) This problem asks for the probability of at least 20 fatalities. Again, it involves counting the number of occurrences, so we need to use the Poisson distribution.
3.66:
a) This problem asks for the probability that at least one sample contains one or more bacteria colonies. Since it involves the occurrence of at least one success, we need to use the complementary probability approach with the Poisson distribution.
b) This problem asks for the number of samples needed to establish a specific probability. We can use the complementary probability approach with the Poisson distribution to find the number of samples needed.
In summary, determining which discrete distribution to use depends on the characteristics of the problem, such as the number of trials, probability of success, whether there is a fixed number of successes, and the need for conditional probabilities or counting occurrences. Understanding these distributions and their properties will help you determine the appropriate one to use for different scenarios.