5.3.2 The Derived Distributions: Student’s t and Snedecor’s F

5.3.2 The Derived Distributions: Student’s t and Snedecor’s F

Definition Let X1, . . . , Xn be a random sample from a N(µ, σ²) distribution. The quantity ( ¯X − µ)/(S/√

n) has Student’s t distribution with n − 1 degrees of freedom. Equivalently, a random variable T has Student’s t distribution with p degrees of freedom, and we write T ∼ t_p if it has pdf

f_T(t) = Γ(^p+1₂ ) Γ(^p₂)

1 (pπ)^1/2

1

(1 + t²/p)^(p+1)/2, −∞ < t < ∞.

Notice that if p = 1, then f_T(t) becomes the pdf of the Cauchy distribution, which occurs for samples of size 2.

The derivation of the t pdf is straightforward. Let U ∼ N(0, 1), and V ∼ χ²_p. If they are independent, the joint pdf is

f_U,V(u, v) = 1

√2πe^−u2/2 1

Γ(p/2)2^p/2v^p2−1e^−v/2, −∞ < u < ∞, 0 < v < ∞.

Make the transformation

t = u

pv/p, w = v, and integrate out w, we can get the marginal pdf of t.

Student’s t has no mgf because it does not have moments of all orders. In fact, if there are p degrees of freedom, then there are only p − 1 moments. It is easy to check that

ETp = 0, if p > 1, VarT_p = p

p − 2, if p > 2.

Example Let X1, . . . , Xnbe a random sample from N(µX, σ_X²) population, and let Y1, . . . , Ym

be a random sample from an independent N(µ_Y, σ_Y²) population. If we were interested in comparing the variability of the populations, one quantity of interest would be the ratio σ²_X/σ²_Y. Information about this ratio is contained in S_X² /S_Y², the ratio of sample variances.

The F distribution allows us to compare these quantities by giving us a distribution of S_X²/S_Y²

σ_X²/σ²_Y = S_X²/σ_X² S_Y²/σ_Y² . 1

Definition Let X₁, . . . , X_nbe a random sample from N(µ_X, σ²_X) population, and let Y₁, . . . , Y_m be a random sample from an independent N(µ_Y, σ_Y²) population. The random variable F = ^S_S^X22^/σX2

Y/σ_Y² has Snedecor’s F distribution with n − 1 and m − 1 degrees of freedom. Equiv- alently, the random variable F has the F distribution with p and q degrees of freedom if it has pdf

f_F(x) = Γ(^p+q₂ ) Γ(^p₂)Γ(^q₂)

¡p q

¢_p/2 x^(p/2)−1

[1 + (p/q)x]^(p+q)/2, 0 < x < ∞.

A variance ratio may have an F distribution even if the parent populations are not normal.

Kelker (1970) has shown that as long as the parent populations have a certain type of symmetric, then the variance ratio will have an F distribution.

Example To see how the F distribution may be used for inference about the true ratio of pop- ulation variances, consider the following. The quantity ^S_S^X22^/σX2

Y/σ_Y² has an F_n−1,m−1 distribution.

We can calculate

EFn−1,m−1 = E¡ χ²_n−1/(n − 1) χ²_m−1/(m − 1)

¢

= E(χ²_n−1/(n − 1))E((m − 1)/(χ²_m−1)) = (m − 1)/(m − 3).

Note this last expression is finite and positive only if m > 3. Removing expectations, we have for reasonably large m,

S_X²/S_Y²

σ_X²/σ_Y² ≈ m − 1 m − 3 ≈ 1, as we might expect.

The F distribution has many interesting properties and is related to a number of other distributions.

Theorem 5.3.8

a. If X ∼ F_p,q, then 1/X ∼ F_q,p; that is, the reciprocal of an F random variable is again an F random variable.

b. If X ∼ tq, then X² ∼ F1,q.

c. If X ∼ F_p,q, then (p/q)X/(1 + (p/q)X) ∼ beta(p/2, q/2).

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