chapter 6: continuous random variables

Question 1

what is the use of continuous probability distributions?

Answer 1

used to find probabilities concerning continuous variable

Question 2

what are three important continuous distributions?

Answer 2

uniform distribution

normal probability distribution

exponential distribution

Question 3

what is a continuous random variable

Answer 3

when a random variable can assume any numerical value in one or more intervals on the real number line

Question 4

how does the continuous probability distribution work?

Answer 4

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

Question 5

normal probability distribution (normal curve)

Answer 5

one continuous probability distribution that graphs as a symmetrical bell shaped curve

Question 6

what are some. of the properties for the continuous probability distribution

Answer 6

1. 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

Question 7

the probability that a continuous random variable x will equal a single numerical value is ….?

why?

Answer 7

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 )

Question 8

when is it reasonable to use the continuous uniform distribution

Answer 8

the relative frequencies between each value of x in the interval is about the same

curve must be straight

overall shape must be rectangular

Question 9

what does the height of the curve f(x) represent?

Answer 9

the relative frequencies of the possible values of x

Question 10

what is the equation of the continuous uniform distribution (uniform model)

Answer 10

f(x) = {1 / (d - c)
{0

c < or = x < or = b

base = (d - c)

height = 1 / (d - c)

Question 11

what is the equation of the area, mean and standard deviation of the continuous uniform distribution (uniform model)

Answer 11

height * base = 1

u = (c + d) / 2

SD = (d - c) / Sqrt(12)

Question 12

what is the equation of the probability that x will be in between the values of the intervals using the uniform distribution

Answer 12

base of rectangle representing the area of the intervals:

(b - a)

P ( a < x < b ) = (b - a) * (1 / (d - c)) = (b - a) / (d - c)

Question 13

what do we use to interpret our calculations with the normal probability distribution (normal model)

Answer 13

the cumulative normal table

for the z scores

Question 14

what are the 5 probabilities of the normal model

Answer 14

1. shape of each normal distribution is determined by its mean and its standard deviation
2. the highest point on the normal curve is located at the mean

    which is  the median and mode of distribution 

3. the curve is symmetrical
4. tails of the curve tend all the way to infinity

   total area under curve is st

5. the area under the curve at the right of the mean equals the area under the curve at the left of the mean

Question 15

if two different curves have the same mean but not the same standard deviation, which one is the flatter?

in normal model

Answer 15

the one with the bigger standard deviation

Question 16

if two different curves have equal standard deviations but unequal means, which one has the mean further to the right?

in normal model

Answer 16

the one with with the biggest mean

Question 17

what are the areas that form the basis of the empirical rule

in normal model

Answer 17

0. 6826 of all observed values are within one standard deviation of the the mean
1. 9544 of all observed values are within two standard deviations of the the mean
2. 9973 of all observed values are within three standard deviations of the the mean

Question 18

true or false

there is a unique curve for every combination of the mean and standard deviation

why?

normal model

Answer 18

true

there are technically unlimited combinations

Question 19

what is the calculation of the z-score (standard normal distribution)

what is the meaning of such calculation

Answer 19

( 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

Question 20

does the z score apply to all normal curves

Answer 20

yeeeeee

Question 21

what is the right hand tail area

Answer 21

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

Question 22

according to z scores, P = 0.9500 is within how many standard deviations of the mean

Answer 22

+ or - 1.96