(Linearity of expectation I) Let X be a random variable, and a, b be constants. Use properties ofintegration/summation to show that:E(aX + b) = aE[X + b]Consider both the discrete and continuous cases.

The linearity of expectation states that the expected value of the sum of random variables is equal to the sum of their individual expected values, regardless of whether they are independent.

Let's consider both the discrete and continuous cases:

1. Discrete Case: Let X be a discrete random variable with a probability mass function p(x). Then the expected value E(aX + b) is given by:

   E(aX + b) = Σ (ax + b)p(x) = aΣ xp(x) + b*Σ p(x) = aE[X] + b

   Here, Σ p(x) = 1 as the sum of probabilities for a probability distribution is always 1.

2. Continuous Case: Let X be a continuous random variable with a probability density function f(x). Then the expected value E(aX + b) is given by:

   E(aX + b) = ∫ (ax + b)f(x) dx = a∫ xf(x) dx + b*∫ f(x) dx = aE[X] + b

   Here, ∫ f(x) dx = 1 as the integral of a probability density function over its entire space is always 1.

Therefore, in both cases, we have E(aX + b) = aE[X] + b, which shows the linearity of expectation.