statistics questions studying for mid sems

Question

1 Let X be normally distributed with mean 10. A statistician is trying to work out P(X > 16). They correctly work out that this probability is 0.04 (rounded to 2dp). Which of the following is the variance of this distribution?

2

Let X ∼ U(0, 1). Which one of the following statements is true

E(X^2 ) = 1/12 . E(X^2 ) = 7 /12 . E(X^2 ) = 4 /12 . E(X^2 ) = 0.

3

Let Y ∼ N(2, 3) and put X = 3(Y − 4). Which one of the following statements is true?

X ∼ N(−6, 9). X ∼ N(6, 27). X ∼ N(−6, 27) X ∼ N(6, 9)

Answer 1

1. Z score giving area of 0.96 is 1.76 so 1.76 = (16 - 10) / sd. Find the sd, then square it.
2. Probability of a value from x to x+dx in [0,1] is 1 dx. So to find expectation integrate x^2 1 dx over [0,1] giving 1/3 (or 4/12)
3. If Y ~ N (p,q) and X = aY + b, then X ~ N (ap+b, aq). Here p=2, q=3, a = 3, b=-12