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A discontinuous [finite] number of outcomes.
Steps for Discrete Probability Distribution:
- 1) Define a variable.
- 2) Create a discrete probability distribution
- 3) Determine if the distribution is valid
- 4) Compute the mean or expected value of x
- 5) Compute the variance and standard deviation
Define binomial distribution.
The discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p.
What would x in a discrete probability distribution problem be referred to as?
x is called the discrete random variable.
Discrete Probability Distribution
What are some properties that make a distribution valid?
- 1) Each probability must be a fraction [0 ≤ probability ≤ 1]
- 2) The sum of all probabilities equal up to 1 [∑ probabilities = 1]
The following formula refers to the whole population if the parameter is considered to be discrete/finite (it is used to compute the mean or expected value of x):
μ = ∑x p(x)
The following formula is used to compute the variance and/or standard deviation:
σ² = ∑x² p(x)-μ²
In discrete probability distributions, each event should be considered to be ______________________ of another.
Explain what is being done with this formula:
P(GGG)= p(G) p(G) p(G)
The probability at each stage is being multiplied.
What are the four properties of a binomial experiment?
- 1) The experiment has been repeated more than once
- 2) There must be only two possible outcomes
- 3) Repetitions of the experiment must be independent of each other
- 4) Probabilities must remain constant from trial to trial
To compute the probabilities of a binomial experiment, you must use the binomial formula:
In the formula: P(x=r)=C(n,r)·pr·(1-p)(n-r), what are the p and n values?
The n [number of trials] and p [probability of success] are called the parameters of the Binomial Distribution and must always be specified in a problem in order to work out the problem.[Note: The r can be any number greater than or equal to 0.]
What does the C(n,r) represent?
C(n,r) represents the number of ways to arrange r successes out of n trials.
The combination formula:
- C(n,r)= n!/[r!(n-r)!]
- The (n,r) can be thought of as a coordinate plane pair (x,y). [Note: n!=n(n-1)(n-2)(n-3)...3·2·1; i.e. 0!=1, 1!=1, 2!=2·1=2, 3!=3·2·1=6, 4!=4·3·2·1=24]
What are the two shortcut formulas for a binomial experiment?
Which formula would be best to solve for the binomial number?
Which formula would be best to solve for the variance?
What determines whether an answer is usual?
In order to be considered a usual amount, a number must fall between two standard deviations of a distribution’s mean [μ ± 2σ].
What is continuous probability distribution?
Continuous probability distribution is when a probability distribution has a cumulative distribution function that is continuous. That means, the variable x has no probability of attaining the value r, therefore its probability will have a zero value [P(x=r)=0]. The x in this case is referred to as a continuous random variable.
In reference to a graph, what property of that graph is equivalent to its probability?
A graph’s area/density/thickness can be used to determine the probability of a graph’s component(s).
While a 1)_____________________ graph can be used as a continuous probability distribution, the 2)__________________________ graph is more commonly used by statisticians.
- 1) uniform
- 2) bell-shaped or normal distribution or Gaussian distribution or bell curve or symmetric shape [might want to know these various names]
Bell-shaped curve or normal distribution or Gaussian distribution or bell curve or symmetric shaped curve
What determines whether an amount is usual in a continuous probability distribution?
The amount must be greater than 5%.
This approach is the one approach used in determining the area of the bell-shaped curve in a calculus-based course.
The normal distribution function
Which continuous probability distribution graph does Z distribution refer to?
It refers to the standard normal distribution.
What do the numbers found in the Z-score table refer to?
The numbers found in the Z-score table refer to the probabilities to the left of a number found on a normal distribution graph.
In the continuous probability distribution, what two properties are relevant?
- 1) The sum of all probabilities must equal 1 [∑ probabilities=1]
- 2) Zero is always in the middle of a standard normal distribution
What formula should be used when converting a scale to the Z-scale?
x has a binomial distribution with the parameters 1)_________ and 2)__________.
- 1) n
- 2) p
- [Note: It is written in (n,p) notation.]
The y has a binomial distribution that differs in its parameters from the x distribution. What are its parameters? (Write it in the correct notation.)
The variables/parameters used in statistics are referred to as ________________.
What two functions best normalize a distribution graph?
- 1) taking the square root of x
- 2) log x
What are the six graphs viewed while assessing normality?
- 1) Histogram
- 2) Stem and leaf
- 3) P-P plots
- 4) Q-Q plots
- 5) Skewness
- 6) Kurtosis
Of the six graphs of assessing normality, which two are used for standard normal distributions?
The P-P plots and Q-Q plots are used when assessing the normality of a standard normal distribution plot.
Of the six graphs viewed while assessing normality, which of the six is used to assess the normality of a numerical quantitative distribution?
Kurtosis is used when assessing the normality of a numerical quantitative distribution.
What is the Central Limit Theorem?
The central limit theorem states that the mean of an independent random variable will be normally distributed [meaning that it will correlate with the standard normal distribution].
Standard deviation _______________ with larger population samples.
If x has a non-normal distribution, then the rule n___30 must be applied.
The formula used to compute probabilities about the mean of x: