Continuous random variable

1
Continuous random variable
The properties of the Probability Density
Function f(x)
2
Continuous random variable
Median and quartiles
½
Q1 is the lower quartile
¼
Q3 is the upper quartile
3
Continuous random variable
The mean and the variance
The expected value x2
Var(x)
4
The Binomial distribution
Conditions
x 0, 1, 2, 3,n
The mean and the variance
The mean ? np
The variance ?2 np(1-p)
5
Approximations
Criteria
n gt ?
50
p close to 0.5 is nice
np gt ? and nq gt ?
5
Continuity correction
Do not forget what ?
np lt 5 or nq lt 5
6
The Poisson distribution
Conditions

• Occur randomly in space or time

• Occur singly

• Occur at a constant rate

• Occur independently

x 0, 1, 2, 3,n
The mean and the variance
The mean The variance
The mean ? ?
The variance ?2 ?
7
The Normal distribution
8
The Normal distribution
Problem that involves finding the value of ?
Example
P(X 20 ) 0.025 and it is given that ? 3.0,
find the value of ?.
0.975
x z
20
? ? 14.1
1.96
9
The Normal distribution
Problem that involves finding the value of ?
Example
P(X lt 10 ) 0.05 and it is given that ? 12.5,
find the value of ?.
0.95
x z
10
? ? 1.52
-1.645
10
The Normal distribution
Problem that involves finding the values of both
? and ?
Example
P(X gt17.24 ) 0.1 and P(X lt14.37)0.25, find the
value of ? and ?.
x z
14.37
17.24
? 15.4 and ? 1.47
-0.674
1.282
11
The Sampling distribution
Unbiased estimate for variance
12
Binomial Hypothesis Testing
Left tail
Right tail
Two tails
13
Poisson Hypothesis Testing
Left tail
Right tail
Two tails
14
The Sampling distribution
Hypothesis testing
15
The Sampling distribution
Testing the mean
Example 10 two tail test