Quiz 3 Flashcards
What is a random variable?
A variable that contains the outcomes of a chance experiment
What is a discrete random variable? Give an example
If the set of all possible values is at most a finite or a countably infinite number of possible values
Determining how many people prefer diet soft drinks
What are continuous distributions?
Give an example
Take on values at every point over a given interval
Measuring the time between customer arrivals at a retail outlet
Discrete and continuous distributions are constructed by what?
Discrete distributions (binomial, Poisson, hypergeometric) are constructed from discrete random variables
Continuous distributions (uniform, normal, exponential, and others) are constructed from continuous random variables
What is the most common graphical way of describing a discrete distribution?
A histogram
Explain the mean of discrete distributions
Mean or expected value
* Defined as the long run average of occurrences
* To think of it in practice, only one trial using discrete values can have 1 outcome, however if it is repeated enough times an expected outcome is created
Explain the variance and standard deviation
The mean has a specific formula and the standard deviation is the square root of the specific formula
What are the assumptions of Binomial distributions?
- The experiment involves n identical trials
- Each trial has only two possible outcomes denoted as success or as failure
- Each trial is independent of the previous trials
- The terms p and q remain constant throughout the experiment, where the term p is the probability of getting a success on any one trial and the term q = (1 − p) is the probability of getting a failure on any one trial
What are binomial distributions characterized by?
Characterized by the parameters n (the sample size) and p (the probability of success)
Explain how to solve a binomial distribution?
It is possible to use the multiplication rule for independent events, however , using the combination rule is easier. By multiplying the combinations by the probability of each gives a binomial formula
When using a binomial table with n = 20 and p= 0.6, what do you summarize if you’re looking for the probability of less than 10 successes?
You add all the probabilities that are less than 10 at the probability of 0.60.
Explain using a computer to produce a binomial distribution
Excel will print binomial table values or find a binomial probability
Explain the mean and standard deviation for binomial distribution
The mean is the sample size multiplied by the probability
The standard deviation is the square root of the sample size multiplied by the probability multiplied by the complement of the probability.
Explain the graphs of the binomial distributions
The mean or the expected long-run average is where the most significant bar graph lies in the histogram, refer to the slides for examples. When the probability is 0.50 the graph is symmetrical. If the probability is higher than 0.50 the graph is skewed left and if the probability is lower than 0.50 the graph is skewed right
What is the Poisson distribution? Give an example
Poisson Probability Distribution:
- A discrete random variable that is often useful in estimating the number of occurrences of an event over a specified interval of time and space.
The Poisson distribution describes the occurrence of rare events
Number of patients who arrive at a health care clinic in one hour.
Number of computer-server failures in a month.
What are the assumptions of the Poisson distribution?
- It is a discrete distribution
- Each occurrence is independent of the other
occurrences - It describes discrete occurrences over a continuum or interval
- The occurrences in each interval can range from zero to infinity
- The expected number of occurrences must hold constant throughout the experiment
Explain the terminologies of Poisson distribution formula
If a Poisson-distributed phenomenon is studied over a long period of time, a long-run average can be determined, λ
where x = 0, 1, 2, 3,… λ = long-run average e = 2.718282
The λ value must hold constant throughout a Poisson experiment
o The analyst must be careful not to apply a given lambda to intervals for which lambda changes
o For example, the average number of customers arriving at a Macy’s store during a one-minute interval will vary from hour to hour, day to day, and month to month
What is the mean and standard deviation of the Poisson distribution?
- The mean of a Poisson distribution is λ, the long-run average of occurrences
- The variance is also λ, so the standard deviation is the square root of À.
When graphing a Poisson distribution what impacts it?
The height and skew of the distribution are determined by λ
What two things must occur for the Poisson distribution to be used with binomial distributions?
- Using the Poisson distribution can approximate certain types of binomial distribution problems.
- Binomial problems that have large sample sizes and small values of p, which then generate rare events, are potential candidates for use of the Poisson distribution.
- If n > 20 and n*p < 7, the approximation can be used
Explain the similarities and differences between the hypergeometric distribution and the binomial distribution
- Like the binomial distribution, the hypergeometric distribution has two outcomes, success or failure
- Unlike the binomial distribution, the analyst must know the size of the population and the probability of success in the population
- Should be used instead of binomial when sampling is done without replacement and the sample is greater than or equal to 5% of the population.
What are the characteristics of the Hypergeometric distribution?
- It is a discrete distribution
- Each outcome consists of either a success or a failure
- Sampling is done without replacement
- The population, N, is finite and known
- The number of successes in the population, A, is known
What is a uniform distribution?
Uniform Distribution
- A relatively simple continuous distribution in which the same height, or f(x), is obtained over a range of values.
How do you determine the probabilities in a uniform distribution?
- For continuous distributions, probabilities are calculated by determining the area over an interval of the function. With continuous distributions, there is no area under the curve for a single point.
What is a normal distribution?
- A widely known and much used continuous distribution that fits the measurements of many human characteristics and many machine-produced items.
What are the characteristics of the normal distribution?
o It is a continuous distribution
o It is a symmetrical distribution about its mean
o It is asymptotic to the horizontal axis
o It is unimodal
o It is a family of curves
o The area under the curve is 1
What are graphical characteristics of the normal distribution?
- The normal distribution is symmetrical. Each half of the distribution is a mirror image of the other half. Many normal distribution tables contain probability values for only one side of the distribution because probability values for the other side of the distribution are identical are identical due to symmetry.
- In theory, the normal distribution is asymptotic to the horizontal axis, meaning it does not touch the x-axis and approaches infinity and negative infinity
- Sometimes referred to as the bell-shaped curve
- It is unimodal in that values mound up in only one portion of the graph – the centre of the curve
- The total area under any normal distribution is 1
The normal distribution is described or characterized by what?
two parameters:
The mean and the standard deviation
What mechanism converts all normal distributions into a single distribution
the z distribution
Explain in depth the z distribution
A normal distribution with a mean of 0 and a standard deviation of 1.
Any value of x at the mean of a normal curve is 0 standard deviations from the mean
Any value of x that is one standard deviation above the mean has a z value of 1
In a z distribution, about 68% of the z values are between z = -1 and z = 1
What is needed to solve Normal curve problems?
Mean,
standard deviation,
z formula
z score table
Explain using the Normal Curve to approximate Binomial distribution problems
- For certain types of binomial distribution problems, the normal distribution can be used to approximate the probabilities
- As sample sizes become large, binomial distributions approach the normal distribution in shape, regardless of the value of p.
- The normal distribution is a good approximation for binomial distribution problems for large values of n
What is required to work on a binomial problem by the normal curves
A translation process:
The first part of this process is to convert the two parameters of a binomial distribution, n and p, to the two parameters of the normal distribution, the mean and standard deviation
After the translation process what test is needed?
A test must be made to determine whether the normal distribution is a good enough approximation of the binomial distribution
Does the interval +- 3 lie between 0 and n?
What are the guidelines for a normal curve approximation of a binomial distribution problem to be acceptable?
All possible x values should be between 0 and n, which are the lower and upper limits, respectively of a binomial distribution
- Another guideline for determining when to use the normal curve to approximate a binomial problem is that the approximation is good enough if both np > 5 and nq > 5
Explain correcting for continuity?
- The translation of a discrete distribution to a continuous distribution is not completely straightforward
- A correction of +- 0.50 is required depending on the problem
- This process ensures that most of the binomial problem’s information is correctly transferred to the normal curve analysis
- This is called correction for continuity, which is made during conversion of a discrete distribution into a continuous distribution
- The decision as to how to correct for continuity depends on the equality sign and the direction of the desired outcomes of the binomial distribution
What is an exponential distribution?
- A continuous distribution that describes a probability distribution of the times between random occurrences.
What are the characteristics or an exponential distribution?
- It is a continuous distribution
- It is a family of distributions
- It is skewed to the right
- The x values range from 0 to infinity
- Its apex is always at x = 0
- The curve steadily decreases as x gets larger
