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 data and the problem you are trying to solve. Here are some guidelines to help you choose the correct distribution:
1. Bernoulli Distribution:
- Use this distribution when you have only two possible outcomes (success or failure).
- The probability of success (p) is constant for each trial.
- Each trial is independent of each other.
2. Binomial Distribution:
- Use this distribution when you have a fixed number of independent trials, and each trial can result in success or failure.
- The probability of success (p) is constant for each trial.
- The number of successes follows a binomial distribution.
3. Discrete Uniform Distribution:
- Use this distribution when all possible outcomes have the same probability of occurring.
- Each outcome is equally likely.
- The number of outcomes is finite.
4. Geometric Distribution:
- Use this distribution when you are interested in the number of trials needed to achieve the first success.
- The probability of success (p) is constant for each trial.
- Each trial is independent of each other.
5. Negative Binomial Distribution:
- Use this distribution when you are interested in the number of trials needed to achieve a specific number of successes.
- The probability of success (p) is constant for each trial.
- Each trial is independent of each other.
6. Poisson Distribution:
- Use this distribution when you are interested in the number of events occurring in a fixed interval of time or space.
- The events occur randomly and independently.
- The average rate of events occurring is known.
Now, let's proceed with answering the questions step-by-step:
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3.40:
a) P(Y >= 2) if p = 0.1 (Geometric Distribution):
- The probability of getting the first success on the k-th trial in a geometric distribution is given by P(Y = k) = (1-p)^(k-1) * p.
- To find P(Y >= 2) in this case, we need to sum the probabilities of getting the first success on the 2nd trial and onwards: P(Y >= 2) = P(Y = 2) + P(Y = 3) + ...
b) P(Y > 4 | Y > 2) for general p (Geometric Distribution):
- This is the conditional probability of getting the first success on the (k+2)-th trial, given that the first success occurred after the 2nd trial.
- We can find it using the conditional probability formula: P(Y > 4 | Y > 2) = P(Y > 4 and Y > 2) / P(Y > 2).
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3.52:
a) Probability of completing the call on the first try (Discrete Uniform Distribution):
- Since there are multiple telephone lines, each with an equal chance of being available or occupied, the probability of completing the call on the first try is 1/(number of telephone lines).
b) Probability that it will take a total of four tries for the two of you (Negative Binomial Distribution):
- This is the probability of getting a total of 4 successes (completed calls) in a sequence of trials until both you and your friend have completed your calls.
- Use the negative binomial distribution formula: P(X = k) = (k-1) C (r-1) * (p^r) * ((1-p)^(k-r)), where X is the number of trials required, k is the total number of trials, r is the number of successes, and p is the probability of success on each trial.
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3.58:
a) Probability that at most 15 fatalities occurred in a given block of 10^9 vehicle miles (Poisson Distribution):
- Use the Poisson distribution formula: P(X <= x) = e^(-λ) * Σ(λ^k / k!), where X is the number of events, x is the maximum number of events, and λ is the average rate of events (fatality rate for this question).
b) Probability that at least 20 fatalities occurred in a given block of 10^9 vehicle miles (Poisson Distribution):
- Use the complement rule: P(X >= x) = 1 - P(X < x), where X is the number of events, x is the minimum number of events, and P(X < x) can be calculated using the Poisson distribution formula.
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3.66:
a) Probability that at least one sample will contain one or more bacteria colonies (Poisson Distribution):
- Use the complement rule: P(X >= 1) = 1 - P(X = 0), where X is the number of samples and P(X = 0) can be calculated using the Poisson distribution formula.
b) Number of 1-cubic-centimeter samples should be selected to establish a probability of approximately 0.95 of containing at least one bacteria colony (Poisson Distribution):
- Use the inverse complement rule: Find the value of X such that P(X >= 1) = 0.95, where X is the number of samples and P(X >= 1) can be calculated using the Poisson distribution formula.
To determine which discrete distribution to use, you need to consider the characteristics of the scenario or problem you are trying to model. Here's a brief explanation of each of the mentioned discrete distributions and when they are commonly used:
1. Bernoulli Distribution: This distribution models a binary event with two possible outcomes, usually referred to as success (1) or failure (0). It is suitable for situations with a single trial, such as flipping a coin or testing the success of a single event.
2. Binomial Distribution: This distribution models the number of successes in a fixed number of independent Bernoulli trials. It is suitable for situations where there are a fixed number of trials, each with the same probability of success, such as the number of heads in multiple coin flips.
3. Discrete Uniform Distribution: This distribution models a situation where each outcome has an equal probability of occurring. It is suitable when there are a finite number of equally likely outcomes, such as rolling a fair die.
4. Geometric Distribution: This distribution models the number of trials needed to achieve the first success in a sequence of independent Bernoulli trials. It is suitable when you want to know the probability of achieving the first success after a certain number of trials.
5. Negative Binomial Distribution: This distribution models the number of trials needed to achieve a fixed number of successes in a sequence of independent Bernoulli trials. It is suitable when you want to know the probability of achieving a specific number of successes after a certain number of trials.
6. Poisson Distribution: This distribution models the number of events that occur in a fixed interval of time or space, where the events occur with a constant mean rate. It is suitable for situations where the events are rare and independent of each other, such as the number of customers arriving at a store or the number of accidents on a road.
Now, taking the given problems one by one, I will explain which discrete distribution should be used to solve them.
3.40:
- This problem involves finding the probability of a geometric distribution, which models the number of trials needed to achieve the first success. Therefore, the geometric distribution should be used.
3.52:
- This problem involves finding the probability of completing a call after a certain number of tries, which suggests a binomial distribution. Each try can be considered a Bernoulli trial with a certain probability of success. The number of tries needed to complete the call follows a binomial distribution.
3.58:
- This problem involves finding the probability of a certain number of fatalities occurring in a given block of vehicle miles. Since the number of events (fatalities) occurring in a fixed interval of time or space is being modeled, a Poisson distribution should be used.
3.66:
- This problem involves finding the probability of the number of bacteria colonies in a sample. Since the number of colonies in a sample follows a Poisson distribution with a given mean, a Poisson distribution should be used.
By understanding the characteristics and appropriate uses of each discrete distribution, you can determine which one to use when faced with a specific problem.