Let Z ∼ Normal(0 , 1) and Y ∼ χ 2 γ , then the new...

Let Z ∼ Normal(0 , 1) and Y ∼ χ 2 γ , then the new r.v. T = √ Z Y /γ has the Student’s t-distribution. The density function of T is fT (t) = Γ[(γ + 1)/2] √γπΓ(γ/2) 1 + t 2 γ !−(γ+1)/2 .

(a) (3 points) Describe the similarity/difference between T and Z.

(b) (6 points) Let t0 be a particular value of t. Use t-distribution table to find t0 values such that the following statements are true. (Given that df=10.) (i) P(−t0 < t < t0) = 0.95. (ii) P(t ≤ −t0 or t > t0) = 0.05. (iii) P(t < t0) = 0.025.

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orchestra answered 1 year ago

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