Download Chapter 6: Probability Distributions Section 1: Random Variables

Chapter 6: Probability Distributions
Section 1: Random Variables and their Distributions
Example: Toss a coin twice
Outcome
# of heads
Probability
HH
2
1/ 4
HT
1
1/ 4
TH
1
1/ 4
TT
0
1/ 4
Definition: A random variable X is a numerical valued function defined on a sample space. It
assigns one numerical value to each point in the sample space.
A discrete random variable is a random variable whose possible values form a finite or countable
infinite set of numbers.
Example: Number of heads in tossing two coins.
A continuous random variable is a random variable whose possible values form a continuum or
interval of numbers.
Example: Computer time in seconds required to process a certain program.
We use capital letters to denote a random variable and lower case letters for their values.
The distribution of a random variable is TWO things:
1. a list of the possible values it can take on;
2. a list of the probabilities of those possible values.
Example 1: For tossing a coin twice. Suppose the random variable X denotes to number of heads.
Write the probability distribution of X.
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Remark: For any discrete distribution, the sum of all probabilities must be 1. i.e.
∑ P( x) = 1.
x
Example 2: For rolling a die twice, if the random variable X denote to the sum of the two faces.
a) Write all possible values of X
b) Write the probability distribution of X
Example 3: If the following table shows the probability distribution for the number of sibling for
a person chosen at random.
x: number of siblings
P(x)
0
0.2
1
2
3
4
0.425
0.275
a
0.025
Find the value of a.
Mean and standard deviation of a Random Variable:
(X )
Definition: The mean or expected value of X =
is µ E=
∑ x P( x) (summation of x times
P(x)).
Definition: The standard deviation of X=
is σ
∑ (x − µ)
2
P( x) .
x
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Example: Find the mean and the standard deviation for the probability distribution in example 1.
Example: Find the mean and the standard deviation for example 2.
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Section 2: The Binomial Random Variable
A binomial experiment is one yielding exactly one of two possible outcomes “success” and
“failure”.
Properties of a binomial experiment
•
There are n independent trials; each one results in either a success (S) or failure (F).
•
The probability of a success, p, remains constant over trials.
•
The random variable of interest is X, the number of successes in n trials.
Definition: A random variable X is said to have a binomial distribution if
n
P( x) =   p x (1 − p ) n − x , x = 0,1, 2,..., n
 x
Where n: number of trials and p: the probability of success.
Use the calculator to get this value: 2ND
VARS
binompdf (n, p, x)
Enter
Example 4: In tossing a coin 5 times, if the random variable X denotes to the number of heads.
Answer the following questions:
a) What are the values of n and p?
b) Find the probability that you get 3 heads?
c) Find the probability of getting no head?
d) Write the probability distribution of X.
e) What is the probability that at least 2 heads occur?
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Example 5: There are 10 senators, each of whom shows for the senate meeting with probability
0.95.
a) What is the probability that no one shows?
b) What is the probability that everyone shows?
c) What is the probability that exactly 7 show?
d) What is the probability that at least 8 show?
Example 6: Sara guesses randomly at 5 multiple choice questions on an exam. Each question has
four potential answers.
a) What are n and p for this question?
b) Find the probability of Sara answers 4 correct?
c) What is the probability of getting all of them correct?
d) Write the probability distribution for this question.
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The mean and standard deviation of a binomial random variable X are
µ = np
•
Mean:
•
Standard deviation:
=
σ
np(1 − p)
Example: Find the mean and the standard deviation for the probability distribution in example 4.
Example: Find the mean and the standard deviation for the probability distribution in example 5.
Example: Find the mean and the standard deviation for the probability distribution in example 6.
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Section 3: Normal Random Variables
Definition: A random variable X is said to have a normal distribution if its probability density
function is given by
f ( x) =
1
σ 2π
e
1  x−µ 2
− 

2 σ 
The mean and standard deviation of X are µ and σ . They both characterize the normal density.
The graph of the density function is
Normal random variables arise in the study of natural and industrial systems.
Many statistical tests are based on the normal distributions.
If X has a normal distribution with mean µ and standard deviation σ . Then P (a ≤ X ≤ b) is the
area of the region under the normal curve between the points a and b above the x-axis:
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•
To find the probability P(a ≤ X ≤ b) where X is the normal random variable with mean
µ and standard deviation σ , use the calculator as P(a ≤ X ≤ b) = normalcdf(a,b, µ , σ ).
To do that use the following commands:
2ND
VARS
normalcdf(a,b, µ , σ )
Enter
Example: The amount of tea X your statistics professor drinks each day has a normal distribution
with mean 55 oz and standard deviation 6 oz. Shade the appropriate regions beneath the density
function and then find the following probabilities:
a) P(49 ≤ X ≤ 55)
b) P(49 < X < 55)
c) P(55 ≤ X ≤ 61)
d) P( X ≤ 55)
e) P( X > 65)
f)
P ( X < 45 or X > 65)
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Example: The score on a national achievement test were approximately normally distributed with
mean 540 and standard deviation 110.
a) If you achieved a score of 680, what percentage of those who took the examination
scored higher than you?
b) What percentage of those who took the examination scored between 300 and 600?
Empirical Rule
If X has a normal distribution with mean µ and standard deviation σ , then
•
68.27% of all observations are within one standard deviation of the mean.
•
95.45% of all observations are within two standard deviations of the mean.
•
99.73% of all observations are within three standard deviations of the mean.
Example: The heights of men in the United States are distributed approximately normal with
mean 69 inches and standard deviation 2.8 inches. Then the empirical rule tells us
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Standard Normal Random Variables:
A standard normal random variable Z has a normal distribution with mean 0 and standard
deviation 1.
i.e. Z has a normal distribution with µ = 0 and σ = 1. The graph of the standard normal
distribution looks like
To find probabilities based on standard normal distribution we can use
P(a ≤ Z ≤ b) = normalcdf(a, b, 0,1).
Example: Find the following
a) P( -2.50 < Z < -1.17)
b) P( -1.20 < Z < 2.40)
c) P( Z < 1.5 )
d) P( Z > -2.42)
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Relationship between Standard Normal Distribution and Normal Distribution
If X is distributed normal with mean µ and standard deviation σ , then Z =
X −µ
σ
has a
standard normal distribution (i.e. µ =0 and σ =1).
Recall that: Z =
X −µ
σ
is the Z-score of X and that means always the Z-score is distributed as
standard normal distribution.
Computing percentile:
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Example: If the score of first exam for 1530 class is normally distributed with mean 70 and
standard deviation 14.
a) Find the score at the 90th percentile?
b) Find the score at the 75th percentile?
c) What score that 95% of students scoreless than it?
d) Find the score that only 2% of students score higher than it?
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