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Determine distribution of aX+b when X~N(0,1)

UpstartData Scientist · HR Screen
EasyData ScientistHR ScreenStatistics & Math

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This problem tests your understanding of how linear (affine) transformations affect the properties of a normal distribution and the derivation of its mean and variance.

Problem Statement

Let X be a random variable that follows a standard normal distribution, denoted as X ~ N(0, 1). Consider a new random variable Y defined by the affine transformation Y = aX + b, where 'a' and 'b' are real-valued constants and 'a' is non-zero.

  1. Determine the mean and variance of Y in terms of 'a' and 'b'.
  2. Prove or explain why Y also follows a normal distribution.
  3. Derive the probability density function (PDF) of Y, denoted f_Y(y), using the change of variables formula for continuous random variables.

Constraints

  • X ~ N(0, 1)
  • a is a non-zero real constant
  • b is any real constant

Example

If a = 2 and b = 5, then Y = 2X + 5.

  • The mean E[Y] would be 2 * E[X] + 5 = 5.
  • The variance Var(Y) would be a^2 * Var(X) = 4 * 1 = 4.
  • Thus, Y ~ N(5, 4).

Hints

Hint 1

Recall the linearity of expectation: E[aX + b] = a*E[X] + b.

Hint 2

Remember that the variance of a constant times a random variable is the constant squared times the variance: Var(aX + b) = a^2 * Var(X).

Hint 3

Use the transformation formula f_Y(y) = f_X(x) * |dx/dy| where x = (y - b) / a.

Follow-up questions

  • How does the distribution of Y change if we define Y = |aX + b| instead?
  • If we have two independent variables X1 ~ N(0, 1) and X2 ~ N(0, 1), what is the distribution of X1 + X2?
  • What would the distribution of Y be if 'a' were allowed to be zero?