Exercises – Section 3.10

Exercises
3.29

Answer the following questions. If the answer is some nonstandard distribution or cannot be determined based upon the information provided, say so.

  1. If N ~ N(1,4), how is M = 3N + 5 distributed?
  2. If L ~ Λ(1,3), how is G = log(L) distributed?
  3. If N ~ N(2,6), how is E = eN distributed?
  4. Assuming that N1 ~ N(0,9) and N2 ~ N(2,1) have correlation 0.3, how is M = N1 + 3N2 distributed?
  5. Assuming that N1 ~ N(1,1) and N2 ~ N(0,4) are independent, how is M = 2N1 + N2 distributed?
  6. If N ~ N(0,1), how is H = N 2 distributed?
  7. Assuming that N1 ~ N(0,1) and N2 ~ N(0,1) are independent, how is H = + (N2 + 5)2 distributed?
  8. If X ~ χ2(1,0), how is C =  distributed?

Solution

3.30

Suppose Z ~ N(0,1). Use a standard normal table to determine Pr(Z ≤ 1.15).
Solution

3.31

For Z ~ N(0,1), use a standard normal table to determine Pr(Z ≤ –0.51). (Hint: Use the symmetry of the normal distribution to find a solution.)
Solution

3.32

Suppose X ~ N(5,7). Use a standard normal table to determine Pr(X ≤ 8).
Solution

3.33

Suppose X ~ N(2, .09). Use a standard normal table to determine the .90 quantile Φ–1(.90) of X.
Solution

3.34

Suppose X ~ N(100, 36). Use a standard normal table to determine the .15 quantile Φ–1(.15) of X.
Solution

3.35

Suppose X ~ N(μ,σ2). Use a standard normal table to determine the .10, .70, and .80 quantiles of X.
Solution

3.36

For X ~ Λ(1.1, .0625), use a standard normal table to determine Pr(X ≤ .9).
Solution

3.37

Suppose X ~ Λ(1.05, 0.01). Use a standard normal table to determine the .75 quantile Φ–1(.75) of X.
Solution

3.38

Prove that, as m → ∞, the skewness and kurtosis of a B(m, p) distribution converge to those of an N(mp, mp(1 – p)) distribution.
Solution