Download 1. Standard Normal Distribution Questions

1. Standard Normal Distribution Questions
There were three types of standard normal distribution questions in the 5th quiz.
1) Probability that Z is smaller than a specified value
2) Probability that Z is larger than a specified value
3) Probability that Z is between two specified values
.
.
.
Standard normal distribution has a mean value of zero and a standard deviation value of 1. Its shape
is given in the following graph.
Distribution Plot
Normal, Mean=0, StDev=1
0.4
Density
0.3
0.2
0.1
0.0
-3
-2
-1
0
X
1
2
3
All three statements form an area under this curve. Usually, in order to find the area under the curve,
we use integrals. However, the integral of the normal distribution function does not have a simple
algebraic form. Therefore, z values are tabulated and the corresponding probabilities are given in
standard normal distribution tables.
These tables generally have the first digit in the rows, and the second digit in the columns, and inside
the table we see the probability values. An example of standard normal distribution table is given
below.
In the rows, we see numbers .0, .1, .2, ... and in the columns we see numbers .00, .01, .02, ...
For example, in order to find the value that corresponds to 1.13 we first find 1.1 in the rows, and
then we find 0.03 in the columns (note that 1.1+0.03=1.13).
This value is the probability that z is smaller than 1.13. Formally,
Instead, if we want to calculate the probability that z is larger than 1.13, we make use of the fact that
the total area under the curve equals 1. Therefore,
When z values are negative, we make use of the fact that the normal distribution table is
symmetrical. For example, the probability that z is larger than 1.13 is equal to the probability that z is
smaller than -1.7. See the graphs for better understanding.
Distribution Plot
Normal, Mean=0, StDev=1
0.4
Density
0.3
0.2
P(z>1.13)
P(z<-1.13)
0.1
0.1292
0.0
0.1292
-1.13
0
X
It is also valid when the inequality signs change. For example,
1.13
Distribution Plot
Normal, Mean=0, StDev=1
0.4
0.3
Density
0.871
0.2
P(z<1.13)
0.1
0.0
0
X
1.13
Distribution Plot
Normal, Mean=0, StDev=1
0.4
0.3
Density
0.871
0.2
P(z>-1.13)
0.1
0.0
-1.13
0
X
For the area between two specified values, we use subtraction.
For example, if we want to find the area between -1.13 and 1.13 we use
To sum up,
1) Probability that Z is smaller than a specified value
.
o If z is positive, find the value that corresponds to z in the table.
 Example:
o If z is negative, find the value that corresponds to |z| in the table, and subtract it
from 1.
 Example:
2) Probability that Z is larger than a specified value
.
o If z is positive, find the value that corresponds to z in the table, and subtract it from
1.
 Example:
o If z is negative, find the value that corresponds to |z| in the table.
 Example:
3) Probability that Z is between two specified values
.
o Find
 Example:
2. Normal Distribution Questions
For normal distribution questions, we first find the corresponding z values, and then do the
calculations that we do in standard normal distribution questions.
For example,
-
Given that X follows a normal distribution with mean 52 and standard deviation 5, what is
P(50<X<57)?
We first find the corresponding z values for 50 and 57.
Now that we have transformed our normal random variables to standard normal variables, we can
find the corresponding probabilities.
First we find
from the table (Row: 1.0, Column: .00) as 0.8413.
Then, we find
using the value corresponds to 0.4 (Row: .4, Column: .00) as 0.6554 and
subtract it from 1 (1-0.6554=0.3446).
Therefore,
3. Uniform Distribution Questions
There were 2 uniform distribution questions. The first one asked the probability density function
value, and the second one asked the expected value.
X follows a continuous uniform distribution with a minimum value of 63 and a maximum
value of 74. What is the probability density function value at X=69.96?
First, let’s remember the shape of continuous uniform distribution.
Distribution Plot
Uniform, Lower=63, Upper=74
0.09
0.08
0.07
Density
0.06
0.05
0.04
0.03
0.02
0.01
0.00
62
64
66
68
X
70
72
74
It has a rectangular shape which makes the calculations very easy. The minimum value is 63 and the
maximum value is 74. We also know that the area of this rectangle is equal to 1. Therefore, in order
to find the height of the rectangle (which is equal to the probability density function value) we use
the following formula:
Note that the probability density function value for all possible x values are the same. Therefore, it is
not important if the problem asks the pdf value at X=69.96 or X=73.81 or X=65.42. They are all the
same and equal to 0.09.
The second uniform distribution question was about the expected value.
X follows a uniform distribution with a minimum value of 31.6 and a maximum value of 54.4.
What is the expected value of X?
Again, since the shape of this distribution is rectangular, the calculations are very easy. The expected
value is the average value and all the points have the same likelihood of occurrence. Therefore, the
expected value is the middle point (between maximum and minimum values).
4. Matching Questions
5. Poisson and Binomial Distribution Questions
There were two questions for these distributions.
When an unfair coin tossed, it land tails 80% of the time. What is the probability that
when it's tossed 23 times, it will land heads at most twice?
This is clearly a binomial distribution question with parameters p=0.2, n=23 and k=2. You are
expected to find
.
Average number of blackouts is 48 per year in a city. What is the probability that
there will be less than or equal to 2 blackouts in a month?
Again, this is clearly a Poisson distribution question with parameter =48 blackouts/year.
You are expected to find
in this question as well.
Refer to the solution key of the 4th quiz for these questions. Since I gave detailed
explanations there, I will not solve them again.