Chapter 10C: Chi-Square and Student t-Distributions  
MATH/STAT 394: Probability I  
Arman Jahangiri  
University of Washington Department of Mathematics  
Summer 2026  
Arman Jahangiri  
Chapter 10C: Chi-Square and Student t-Distributions  
Summer 2026  
1
 
Learning Goals  
By the end of this lecture, students should be able to:  
▶ define the Chi-Square distribution;  
▶ understand the connection between Chi-Square and Gamma distributions;  
▶ compute means and variances of Chi-Square random variables;  
▶ understand why Chi-Square distributions appear in statistics;  
▶ define the Student t-distribution;  
▶ understand why t-distributions have heavier tails than Normal distributions;  
▶ understand convergence of the t-distribution to the Normal distribution.  
Arman Jahangiri  
Chapter 10C: Chi-Square and Student t-Distributions  
Summer 2026  
2
Chi-Square Distribution  
Arman Jahangiri  
Chapter 10C: Chi-Square and Student t-Distributions  
Summer 2026  
3
Why Chi-Square Distributions Matter  
Chi-Square distributions appear throughout statistics:  
▶ sample variance;  
▶ hypothesis testing;  
▶ confidence intervals;  
▶ goodness-of-fit tests;  
▶ contingency tables.  
Main idea  
Chi-Square distributions measure accumulated squared Normal variation.  
Arman Jahangiri  
Chapter 10C: Chi-Square and Student t-Distributions  
Summer 2026  
4
Definition of the Chi-Square Distribution  
Definition  
Suppose  
Z1, . . . , Zn  
are independent standard Normal random variables.  
Define  
X = Z12 + · · · + Zn2.  
Then  
X ∼ χ2n.  
We say that X has a Chi-Square distribution with  
n
degrees of freedom.  
Arman Jahangiri  
Chapter 10C: Chi-Square and Student t-Distributions  
Summer 2026  
5
Why Squaring Matters  
Each  
Zi2  
measures squared deviation from the mean.  
Adding them gives a total amount of variation:  
Z12 + · · · + Zn2.  
Interpretation  
Chi-Square random variables measure total squared fluctuation.  
Arman Jahangiri  
Chapter 10C: Chi-Square and Student t-Distributions  
Summer 2026  
6
Chi-Square and Gamma  
Recall: if  
then  
X ∼ Gamma(α, λ),  
λα  
f (x) =  
xα−1e−λx  
.
Γ(α)  
A Chi-Square distribution is a special Gamma distribution:  
ꢀ
ꢁ
n 1  
,
χ2n = Gamma  
.
2 2  
Arman Jahangiri  
Chapter 10C: Chi-Square and Student t-Distributions  
Summer 2026  
7
PDF of the Chi-Square Distribution  
Using the Gamma formula:  
1
f (x) =  
xn/2−1e−x/2  
,
x > 0.  
2n/2Γ(n/2)  
The shape depends heavily on the degrees of freedom.  
Arman Jahangiri  
Chapter 10C: Chi-Square and Student t-Distributions  
Summer 2026  
8
Mean and Variance of Chi-Square  
If  
X ∼ χ2n,  
then  
E(X) = n.  
Also,  
Var(X) = 2n.  
Observation  
Both the mean and variance grow linearly with the degrees of freedom.  
Arman Jahangiri  
Chapter 10C: Chi-Square and Student t-Distributions  
Summer 2026  
9
Shape of Chi-Square Distributions  
Small degrees of freedom:  
▶ strongly right-skewed;  
▶ concentrated near zero.  
Large degrees of freedom:  
▶ less skewed;  
▶ more symmetric;  
▶ approximately Normal.  
Connection to CLT  
A Chi-Square distribution becomes approximately Normal for large  
n.  
Arman Jahangiri  
Chapter 10C: Chi-Square and Student t-Distributions  
Summer 2026 10  
Sample Variance and Chi-Square  
Suppose  
i.i.d.  
X1, . . . , Xn ∼ N(µ, σ2).  
Recall the sample variance:  
n
X
1
Sn2  
=
(Xj − Xn) .  
2
¯
n − 1  
j=1  
Then:  
(n − 1)Sn2  
∼ χ2n−1  
.
σ2  
This is one of the most important results in classical statistics.  
Arman Jahangiri  
Chapter 10C: Chi-Square and Student t-Distributions  
Summer 2026 11  
Degrees of Freedom  
Why do we get  
n − 1  
degrees of freedom instead of  
n?  
Because:  
¯
Xn  
has already been estimated from the data.  
Once the sample mean is fixed, only  
n − 1  
deviations can vary independently.  
Arman Jahangiri  
Chapter 10C: Chi-Square and Student t-Distributions  
Summer 2026 12  
Chi-Square Additivity  
Theorem  
If  
X ∼ χ2m,  
Y ∼ χ2n,  
independently, then  
X + Y ∼ χ2m+n  
.
This follows immediately from the Gamma representation.  
Arman Jahangiri  
Chapter 10C: Chi-Square and Student t-Distributions  
Summer 2026 13  
Student t-Distribution  
Arman Jahangiri  
Chapter 10C: Chi-Square and Student t-Distributions  
Summer 2026 14  
Why the t-Distribution?  
Suppose:  
▶ the population is Normal;  
▶ the variance  
σ2  
is unknown;  
▶ the sample size is small.  
Then the standard Normal approximation is no longer exact.  
Solution  
The Student t-distribution accounts for uncertainty in estimating the variance.  
Arman Jahangiri  
Chapter 10C: Chi-Square and Student t-Distributions  
Summer 2026 15  
Definition of the Student t-Distribution  
Definition  
Let  
Z ∼ N(0, 1),  
and  
V ∼ χ2n,  
independently.  
Define  
Z
p
T =  
.
V /n  
Then  
T
has the Student t-distribution with  
degrees of freedom.  
n
Arman Jahangiri  
Chapter 10C: Chi-Square and Student t-Distributions  
Summer 2026 16  
Why the t-Distribution Has Heavy Tails  
The denominator:  
p
V /n  
is random.  
Sometimes it becomes unusually small.  
That makes:  
|T|  
occasionally very large.  
Consequence  
The t-distribution has heavier tails than the Normal distribution.  
Arman Jahangiri  
Chapter 10C: Chi-Square and Student t-Distributions  
Summer 2026 17  
Properties of the t-Distribution  
The t-distribution is:  
▶ symmetric about  
0;  
▶ bell-shaped;  
▶ heavier-tailed than the Normal distribution.  
As the degrees of freedom increase:  
▶ tails become lighter;  
▶ the distribution approaches the Normal distribution.  
Arman Jahangiri  
Chapter 10C: Chi-Square and Student t-Distributions  
Summer 2026 18  
Convergence to the Normal Distribution  
As  
n → ∞,  
the Chi-Square random variable satisfies:  
Therefore:  
V
→ 1.  
n
Z
p
T =  
≈ Z.  
V /n  
Thus:  
tn → N(0, 1).  
Arman Jahangiri  
Chapter 10C: Chi-Square and Student t-Distributions  
Summer 2026 19  
The Cauchy Distribution  
If  
n = 1,  
then the t-distribution becomes the Cauchy distribution.  
The Cauchy distribution has:  
▶ extremely heavy tails;  
▶ undefined mean;  
▶ undefined variance.  
Important  
Not every distribution has finite expectation or variance.  
Arman Jahangiri  
Chapter 10C: Chi-Square and Student t-Distributions  
Summer 2026 20  
Confidence Intervals  
The t-distribution is fundamental for confidence intervals.  
For example:  
Sn  
∗
¯
√
Xn ± t  
.
n
Here:  
▶
¯
Xn  
Sn  
t∗  
is the sample mean;  
▶
estimates the standard deviation;  
▶
comes from the t-distribution.  
Arman Jahangiri  
Chapter 10C: Chi-Square and Student t-Distributions  
Summer 2026 21  
Normal vs t-Distribution  
Distribution Tails  
Variance known?  
yes  
Normal  
lighter  
heavier no  
Student t  
Interpretation  
The heavier tails account for uncertainty in estimating the variance.  
Arman Jahangiri  
Chapter 10C: Chi-Square and Student t-Distributions  
Summer 2026 22  
Big Picture  
Three major distributions now fit together:  
Distribution  
Built From  
Main Role  
Normal  
Chi-Square  
Student t  
sums/CLT  
averages  
variances  
inference  
squared Normals  
Normal divided by Chi-Square  
Arman Jahangiri  
Chapter 10C: Chi-Square and Student t-Distributions  
Summer 2026 23  
Summary  
▶ Chi-Square distributions are sums of squared standard Normals.  
▶ Chi-Square distributions are special Gamma distributions.  
▶ Sample variances naturally produce Chi-Square distributions.  
▶ Student t-distributions arise when variance is estimated.  
▶ t-distributions have heavier tails than Normals.  
▶ As degrees of freedom increase:  
tn → N(0, 1).  
End of Probability I  
The LLN, CLT, Chi-Square, and t-distributions form the mathematical foundation of classical statistics.  
Arman Jahangiri  
Chapter 10C: Chi-Square and Student t-Distributions  
Summer 2026 24