Suppose that the length, in inches, of a witch's or wizard's wand is a normal random variable with parameters mu = 11 and sigma^2 = 2. (a) what percentage of wands are over 12 inches? leave you answer in terms of phi. (b) given that a wand is greater than 12 inches, what is the probability it is greater than or equal to 15 inches? leave you answer in terms of phi.

a)

b)

Step-by-step explanation:

1) Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  The letter is used to denote the cumulative area for a b quantile on the normal standard distribution, or in other words:

Let X the random variable that represent the length of a population, and for this case we know the distribution for X is given by:

Where and

2) Part a

We are interested on this probability

And the best way to solve this problem is using the normal standard distribution and the z score given by:

If we apply this formula to our probability we got this:

And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.  

3) Part b

For this case we want a conditional probability given by:

And in order to solve this probability we can use the Bayes theorem given by:

If we use this rule we have this:

The denominator is already founded on part a. So we need to find:

And the best way to solve this problem is using the normal standard distribution and the z score given by:

If we apply this formula to our probability we got this:

And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.  

the most boring ever written

step-by-step explanation:

i am writing something. yes, i plan to make it the most boring thing ever written. i go to the store. a car is parked. many cars are parked or moving. some are blue. some are tan. they have windows. in the store, there are items for sale. these include such things as soap, detergent, magazines, and lettuce. you can enhance your life with these products. soap can be used for bathing, be it in a bathtub or in a shower. apply the soap to your body and rinse. detergent is used to wash clothes. place your dirty clothes into a washing machine and add some detergent as directed on the box. select the appropriate settings on your washing machine and you should be ready to begin. magazines are stapled reading material made with glossy paper, and they cover a wide variety of topics, ranging from news and politics to business and stock market information. some magazines are concerned with more recreational topics, like sports card collecting or different kinds of hairstyles. lettuce is a vegetable.

no oblique asymptote

step-by-step explanation:

to find the oblique asymptote of f(x)

here numerator has degree 1 and denomintor degree 2

degree of numerator < degree of denominator

hence there is no oblique asymptote

when the degree of the denominator > that of the numerator, we saw that the value in the denominator got so much bigger, so quickly,   giving us a horizontal asymptote of the x-axis. or we can write

f(x) = a polynomial of x + a proper fraction.

in this case only we have oblique asymptotes but not when numerator has less degree.