chapter 6 Flashcards
what are the 3 continuous random variables distributions
- uniform
- normal
- exponential
- normal approximation to the binomial
what is the fundamental different b/w discrete and continuous random variables
how they are computed
Explain Fx for a discrete random variable
provides the prob a random variable assumes a particular value
explain Fx for a continuous random variable
it is the probability density function
- does not directly provide the prob of the continuous random variable x
- it does provide the continuous random variable x assumes a value in the interval
-
for a continuous random variable, the area under th graph of f(x) at any point is
0
for continuous random variables F(x) must be what for all values of x
> or = 0
greater than or equal to zero
for uniform prob distribution - how do you calculate f(x)
1/ (b-a)
for uniform prob distribution - how do you calculate the prob of an interval
(b-a)x F(x)
For uniform prob distribution - how do you calculate expected value (or mean)
(a+b) / 2
for uniform prob distribution - how do you calcualte the variance
(b-a)squared / 12
if x is a continous random variable then, x can assume what
any value in an interval
- intervals are equally likely
F(x) probaility density function (uniform) the area of a rectangle is
width x height
What are the two major differences b/w the treatment of continous random variables and discrete random variables
- we no longer talk about the prob of the random variable assuming a particular value
- we talk about the prob of the random variable assuming a value within some given Interval - the prob of a continuous random variable assuming a value w/in some given interval from x1 to x2 is defined to be the
- the area under the graph of the prob density function b/w x1 and x2
b/c a single point of any interval of 0 width implies what
- that for x to be exactly a number = 0
2. the prob of a x assuming a value in any interval is the same whether or not the end points are included
Is the height of a density function a probability?
NO
provide examples of when to use the normal prob distribution
a wide variety of practical applications
- heights and weights of people
- test scores
- scientific measures
- amounts of rainfall etc
which probability density function is the most important
normal
what is the normal prob distribution used as
a statistical inference where the normal distribution provides a description of the likely results obtained thorough sampling
the normal curve has what shape
bell curve
what are the 6 characteristics of a normal distribution
- has 2 parameters (m - mean and Q - standard deviation)
- the highest point on the curve is at the mean, which is also the median and mode
- the mean can be any numerical value negative, o or positive
- it is symmetric (it goes to infinity in both directions)
- each side from the mean is the mirror image
- skewness is zero - the SD determines how flat and wide the curve is
- larger SD = wider and flatter curves (shows more variability in the data) - Prob(s) are given by areas under the curve
- total area under the curve = 1
What are some commonly used intervals for the normal distribution
- 68.3% of the values of a normal random variable are w/in plus or mins 1 sd of its mean
- 95.4 % 2 SD
- 99.7% 3SD
What are the characteristics of a standard NOrmal Prob Distribution
mean = 0
SD = 1
Z- is used to designate this particular normal random variable (z-scores convert the data to a a standard normal prob distribution so we can use z-tables to make inferences)
what does the z-score tell us
how many sd it is away form the mean
- it allows us to calculate the area the z-score is associated with using z-tables
Standardization uses the formula
z= x-m / SD
Why do we use the normal approximation of the binomial probabilities
- when the # of trials becomes large, evaluating the binomial prob function by hand or with a calculator is difficult
What does a binomial experiment consist of
consists of a sequence of n identical independent trials with each trial having 2 possible outcomes, a success or a failure
with a binomial experiment the prob of success on any trial is
the same for all trials
what is the continuity correction error
.5 is added and/or subtracted to calculate the z-score
For the binomial continuity correction error
if you want know for example 13 or fewer
13.5 is used to compute the z-score
For the binomial continuity correction error
if you want know for example exactly 12
compute z-score for 11.5 and then 12.5 and subtract
Exponential distribution has a mean and sd of what
they are equal or the same
m=sd
for Exponential dist. how do you calculate less than or = 6
1-e to exponent of -6/mean
DOUBLE CHECK THIS ANSWER
for exponential dist. how do you calculate great than 6
calculate less than 6 then 1 - this number
DOUBLE CHECK THIS ANSWER
for exponential dist. how do you calcualte b/w two numbers example b/w 4 and 6
you first calculate using -4/m
then you do it with -6/m and subtract the two
what is the skewness measure for the exponential distribution
2
and is skewed right
how do you calculate the the sd for the exponential
m=Q
so
What are some examples the exponential prob distribution used for?
- time b/w arrivals at a car wash
- time required to load a truck
- distance b/w major defects in a highway etc
what is the relationship b/w exponential and poisson
Poisson
- provides an appropriate description of the # of OCCRURRENCES per interval
Exponential
- provides a description of LENGTH of Interval b/w Occurrences
For any continuous random variable, the probability that the random variable takes a value less than zero is
any number between zero and 1
For the standard normal prob distribution, the area to the right of the mean is more than______.
.50
The prob that a continuous random variable takes on any specific value is
0
a normal distribution with a mean of zero and sd of 1 is called
a standard normal distribution
the z score for the standard normal distribution can be
either negative or positive
in a standard normal distribution, the prob that z is greater than zero is
0.5
a negative value for z indicates that
the number of sd of an observation is to the left of the mean
for a standard normal distribution, the prob of z is greater than zero is
.50 (remember the mean is 0)
