1. Chi-Squared Distribution Leadership in Engineering
Systems Engineering Program
Department of Engineering Management, Information and Systems
EMIS 7370/5370 STAT 5340 :
PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS
Special Continuous Probability Distributions
t- Distribution
Chi-Squared Distribution
F- Distribution
Dr. Jerrell T. Stracener, SAE Fellow
Leadership in Engineering

2. t-Distribution

3. The t-Distribution
Definition - A random variable T is said to have the
t-distribution with parameter , called degrees of
freedom, if its probability density function is given by:
, -  < t < 
where is a positive integer.

4. Values of T, tp,ν for which P(T > tp,ν) = p
The t-Distribution
Remark: The distribution of T is usually called the
Student-t or the t-distribution. It is customary to let
tp represent the t value above which we find an area
equal to p. p tp t
Values of T, tp,ν for which P(T > tp,ν) = p

5. probability density function for various values of 
The t-distribution
probability density function for various values of 

6. Table of t-Distribution
t-table gives values of tp for various values of p
and ν. The areas, p, are the column headings; the
degrees of freedom, ν, are given in the left column,
and the table entries are the t values.
Excel

7. t-Distribution - Example
If T~t10,
find:
(a) P(0.542 < T < 2.359)
(b) P(T < )
(c) t′ for which P(T>t′) =

8. Chi-Squared Distribution

9. The Chi-Squared Distribution
Definition - A random variable X is said to have the
Chi-Squared distribution with parameter ν, called
degrees of freedom, if the probability density function
of X is:
, for x > 0
, elsewhere
where ν is a positive integer.

10. The Chi-Squared Model Remarks:
The Chi-Squared distribution plays a vital role in
statistical inference. It has considerable application
in both methodology and theory. It is an important
component of statistical hypothesis testing and
estimation.
The Chi-Squared distribution is a special case of the
Gamma distribution, i.e., when  = ν/2 and  = 2.

11. The Chi-Squared Model - Properties
Mean or Expected Value
Standard Deviation

12. The Chi-Squared Model - Properties
It is customary to let 2p represent the value above
which we find an area of p. This is illustrated by the
shaded region below. x
f(x) p
For tabulated values of the Chi-Squared distribution see the
Chi-Squared table, which gives values of 2p for various values
of p and ν. The areas, p, are the column headings; the degrees
of freedom, ν, are given in the left column, and the table entries
are the 2 values.
Excel

13. The Chi-Squared Model – Example
If ,
find:
(a) P(7.261 < X < )
(b) P(X < 6.262)
(c) c’ for which P(X < c’) = 0.02 c X ~ 2 15

14. F-Distribution

15. The F-Distribution
Definition - A random variable X is said to have the
F-distribution with parameters ν1 and ν2, called
degrees of freedom, if the probability density function
is given by:
, 0 < x < 
, elsewhere
The probability density function of the F-distribution
depends not only on the two parameters ν1 and ν2
but also on the order in which we state them.

16. The F-Distribution - Application
Remark: The F-distribution is used in two-sample
situations to draw inferences about the population
variances. It is applied to many other types of
problems in which the sample variances are involved.
In fact, the F-distribution is called the variance ratio
distribution.

17. Probability density functions for various values of ν1 and ν2
The F-Distribution
f(x)
6 and 24 d.f.
6 and 10 d.f. x
Probability density functions for various values of ν1 and ν2

18. The F-Distribution - Properties
Table: The fp is the f value above which we find an area
equal to p, illustrated by the shaded area below.
f(x) p x
For tabulated values of the F-distribution see the F
table, which gives values of xp for various values of ν1
and ν2. The degrees of freedom, ν1 and ν2 are the
column and row headings; and the table entries are
the x values.
Excel

19. The F-Distribution - Properties
Let x(ν1, ν2) denote x with ν1 and ν2 degrees of
freedom, then

20. The F-Distribution – Example
If Y ~ F6,11,
find:
(a) P(Y < 3.09)
(b) y’ for which P(Y > y’) = 0.01