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Probability and statistics
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Terms in this set (46)
random variable
a numerical measure of the outcome of an experiment, so its value is determined by chance.
discrete random variable
random variable with either a finite(whole) number value or a countable number.
continous random variable
A random variable that can take any numeric value within a range of values.The range may be infinite or bounded at either or both ends.
probability distribution
list of possible outcomes with associated probabilities. Can be in a form of a table, graph, or mathematical formula.
Discrete probability distribution
the SUM of all Probability of X is equal to 1.
each probability is between 0 and 1 inclusive.
probability histogram
a histogram in which the horizontal axis corresponds to the value of the random variable and the vertical axis represents the probability of each value of the random variable.
distribution of variable
mean-describes the center.
standard deviation-describe the spread.
The mean of a discrete Random variable
by multiplying each possible value of the random variable by its corresponding probability and then add their products.
Interpretation of the mean of the discrete random variable
as the number of n increases, the mean of the observations will approach the mean of the random variable. (the series if played many times would expect to last 5.8 games on average)
Standard deviation of discrete random variable
it is the square root of a weighted average of the squared deviations for which the weights are the probabilities.
binomial probability distribution
The probability distribution for a discrete random variable, used to compute the probability of x successes in n trials. There are two mutually exclusive outcomes-success and failure (binomial experiments)
Binomial probability experiment (criteria)
-it is performed a fixed no. of times (not continous)
-trials are independent.(one trial does not affect the outcome of other trial)
-for each trial- there are two mutually exclusive (disjoint) outcomes (S and P)
-Probability of success is the same each trial of experiment.
trials
each repetition of the experiment.
n (in binomial probability distribution)
independent trials of the experiment.
p (in binomial probability distribution)
the probability of success
1-p
is the probability of failure for each trial.
let X in (0 greater than or equal X less than or equal)
number of successes in n independent trial.
finding probability exactly 10
binompdf
finding probability no morethan 6 (less than or equal to 6)
binomcdf
computing the mean of binomial random variable
Ux= np
computing the standard deviation of binomial random variable
0x= square root of np(1-p)
In binomial distribution
for fixed p, as the number of trials n increases, the probability distribution of the random variable x becomes bell shaped.
As a rule of thumb (binomial dist)
if np(1-p) is greater or equal 10, the probability distribution will be approximately bell shaped. (ex. n=70 p=.2 is 11.2, that is why it is bell shaped)
empirical rule
The rules gives the approximate % of observations w/in 1 standard deviation (68%), 2 standard deviations (95%) and 3 standard deviations (99.7%) of the mean when the histogram is well approx. by a normal curve
checking unusual results
rule: any observation is less than u-2o or greater u+2o is unusual.
uniform probability distribution
a continuous probability distribution for which the probability that the random variable will assume a value in any interval is the same for each interval of equal length.
probability density function
is an equation used to compute probabilities of continuos random variables.
-the total area under the graph of the equation overall possible values must equal 1.
-the height of the graph of the equation must be grater than or equal to 0 for all possible values of the random variable.
normal distribution
A function that represents the distribution of variables as a symmetrical bell-shaped graph.
normal curve (model)
which is used to describe continous random variables that are said to be normally distributed.
normally distributed/ normal probability distribution
if its relative frequency histogram has the shape of a normal curve.
inflection points
The point at which a change of curvature takes place
properties of normal density curve
Symmetry about the mean u, Because mean = median = mode, the curve has a single peak and the highest point occurs at x = μ. It has inflection points at μ - σ and μ - σ. The area under the curve is 1.
The area under the curve to the right of μ equals the area under the curve to the left of μ, which equals 1/2.
As x increases or decreases, the graph approaches but never reaches the horizontal axis.the Empirical rule is applied.
Interpreting the area under a normal curve
the proportion of the population (20-29 year old males that have high cholesterol) with the characteristic described by the interval or values (is 0.2903)
standard normal variable z
A random variable that has a normal distribution with a mean of 0 and a standard deviation of 1.
standard normal curve
A normal distribution with mean of zero and standard deviation of one. Probabilities are given in Table A for values of the standard Normal variable.
complement rule
the probability of an event occurring is 1 minus the probability that it doesn't occur. (finding area to the right)
z score
x-mean(u)/standard deviation.
sampling distribution
a distribution of statistics obtained by selecting all the possible samples of a specific size from a population
sampling distribution of the sample mean
the probability distribution of all possible values of the random variable x̄ computed from a sample size n from a population with mean µ and standard deviation σ
standard error of the mean
the standard deviation of the sampling distribution of the mean (Ox=o/square root of n)
the shape of the sampling distribution of mean if X is normal
if the population is normal, then the distribution of the mean is normal.
the central limit theorem
The larger the sample, the better the approximation will be. as the sample size increases, n , the sampling distribution becomes approximately normal, regardless of the shape of the underlying population.
-has to do with the shape of distribution of mean, not the center of spread.
distribution of the sample mean
-if the population is normal with mean and standard deviation, regardless of the sample size n, the shape of the sample mean is normal.
-if population is not normal with mean and standard deviation, as the sample size n increases, the distribution of sample mean becomes approximately normal.
sample proportion
The proportion (percentage) of a sample from the population that has the specified attribute (denoted p-hat).
p-hat
is given by phat=x/n--where x is the number of individuals in the sample.
-is a statistic that estimates the population proportion p.
sampling distribution of p-hat
-the shape of the sampling distribution of p-hat is approximately normal provided...np(1-p)greater or equal 10.
-the mean of sampling distributionof p-hat=p
-the standard deviation of the sampling distribution of p-hat= square root of p(1-p)/n.
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