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Part CCSIR NET June 2024variance-scales-by-the-square

Variance scales by the square

Let X and Y be independent random variables with X ~ N(2, 4) and Y ~ N(−4, 9) where N(μ,σ2)N(\mu, \sigma^{2}) denotes a normal distribution with mean μ\mu and variance σ2\sigma^{2}. Given Φ(1)=0.8413,Φ(2)=0.9772\Phi(1) = 0.8413, \Phi(2) = 0.9772 and Φ(3)=0.9987\Phi(3) = 0.9987 where Φ(⋅)\Phi(\cdot) is the cumulative distribution function of a standard normal random variable. Which of the following statements are true?

  1. A.Var(2X + Y) = 17
  2. B.P(|2X + Y| ≤ 15) = 0.9974✓ Correct
  3. C.Cov(3X + 2Y, 3X − 2Y) = 0✓ Correct
  4. D.2X − Y ~ N(0, 25)

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests execution slip.

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50 are analysed free — try those first.

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

Related counterexample: Uncorrelated ⇒ independent

More on this topic

The chapter behind this: Random variables, moments and generating functions — free to read

From Probability › Random variables, distributions, moments, MGF

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