chapter 6: continuous random variables Flashcards
what is the use of continuous probability distributions?
used to find probabilities concerning continuous variable
what are three important continuous distributions?
uniform distribution
normal probability distribution
exponential distribution
what is a continuous random variable
when a random variable can assume any numerical value in one or more intervals on the real number line
how does the continuous probability distribution work?
it assigns probabilities to intervals of values
you use a curve f(x) to represent the real number line
the curve F(x) is the continuous probability distribution of the random variable x if:
the probability that x will be in a specified interval of numbers represents the area under the curve f(x) that is equivalent to that interval
normal probability distribution (normal curve)
one continuous probability distribution that graphs as a symmetrical bell shaped curve
what are some. of the properties for the continuous probability distribution
- f(x) must be greater or equal to x for any value of x
2. the total area under the curve must be equal to 1
the probability that a continuous random variable x will equal a single numerical value is ….?
why?
probability is 0
if [ a , b ] denotes arbitrary interval of numbers on the real line,
P( x = a ) and P( x = b ) = 0
then all in all, P( a < x < b )
when is it reasonable to use the continuous uniform distribution
the relative frequencies between each value of x in the interval is about the same
curve must be straight
overall shape must be rectangular
what does the height of the curve f(x) represent?
the relative frequencies of the possible values of x
what is the equation of the continuous uniform distribution (uniform model)
f(x) = {1 / (d - c)
{0
c < or = x < or = b
base = (d - c)
height = 1 / (d - c)
what is the equation of the area, mean and standard deviation of the continuous uniform distribution (uniform model)
height * base = 1
u = (c + d) / 2
SD = (d - c) / Sqrt(12)
what is the equation of the probability that x will be in between the values of the intervals using the uniform distribution
base of rectangle representing the area of the intervals:
(b - a)
P ( a < x < b ) = (b - a) * (1 / (d - c)) = (b - a) / (d - c)
what do we use to interpret our calculations with the normal probability distribution (normal model)
the cumulative normal table
for the z scores
what are the 5 probabilities of the normal model
- shape of each normal distribution is determined by its mean and its standard deviation
- the highest point on the normal curve is located at the mean
which is the median and mode of distribution
- the curve is symmetrical
- tails of the curve tend all the way to infinity
total area under curve is st
- the area under the curve at the right of the mean equals the area under the curve at the left of the mean
if two different curves have the same mean but not the same standard deviation, which one is the flatter?
in normal model
the one with the bigger standard deviation
if two different curves have equal standard deviations but unequal means, which one has the mean further to the right?
in normal model
the one with with the biggest mean
what are the areas that form the basis of the empirical rule
in normal model
- 6826 of all observed values are within one standard deviation of the the mean
- 9544 of all observed values are within two standard deviations of the the mean
- 9973 of all observed values are within three standard deviations of the the mean
true or false
there is a unique curve for every combination of the mean and standard deviation
why?
normal model
true
there are technically unlimited combinations
what is the calculation of the z-score (standard normal distribution)
what is the meaning of such calculation
( x - mean ) / ( standard deviation )
expresses the normal of standard deviations that x is form the mean
is it to find the area under normal curve
does the z score apply to all normal curves
yeeeeee
what is the right hand tail area
it is the portion that we have to calculate like:
P (z > 2) =
1 - P ( z < 2 ) for example
the value of z (2) has to remain positive
according to z scores, P = 0.9500 is within how many standard deviations of the mean
+ or - 1.96
