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**Student’s T-Distribution** (sometimes called the* Student Distribution* or *T-Distribution*) is a family of probability distributions that are similar in shape to the normal distribution. Student’s t-distribution is used instead of the normal distribution when you have small samples or don’t know the population standard deviation. In real life, you usually *don’t* know the population standard deviation, so Student’s t is seen more frequently in real-world practical applications than the normal distribution.

## PDF and CDF

The probability density function is:

And the cumulative distribution function is:

Where

_{2}F_{1 }is the hypergeometric function,- Γ is the gamma function,
- Ν is degrees of freedom.

A related distribution is Hotelling’s distribution, which for *q* = 1 is the positive half of Student’s T; Hotelling’s is sometimes called the *generalized Student *[1]. The standard probability density function of Cauchy distribution is a t-distribution with 1 degree of freedom [2].

Student’s T Distribution is used in hypothesis testing (the “t-test”) to figure out if you should accept or reject the null hypothesis.

## How Student’s T-Distribution got its Name

The distribution was developed by William Gosset under the pseudonym “Student”. Gosset used a pseudonym because the company he worked for—Guiness—prohibited its employees from publishing under their real names due to some trade secret leaks in earlier scientific publications [3].

References

[1] Haight, F. (1958). Index to the Distributions of Mathematical Statistics. National Bureau of Standards Report.

[2] Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.

[3] Josic, K. No. 3072. William Gosset. Online: https://www.uh.edu/engines/epi3072.htm