Expected Value and Variance

These flashcards help summarize and reinforce key concepts related to expected value and variance as discussed in the lecture.

What does the expected value of a random variable indicate?

It tells us the center of mass of a distribution.

How is the expected value of a discrete random variable defined?

E[X] = Σ xi · pX(xi) where xi are the values of the random variable and pX(xi) is the probability mass function.

What is the formula for the variance of a random variable X?

Var(X) = E[(X - E[X])²] or Var(X) = E[X²] - (E[X])².

What does the variance measure?

Variance measures how spread out the values of a random variable are around the mean.

What is the relationship between variance and standard deviation?

The standard deviation is the square root of the variance.

What is the linearity property of expectation in terms of two random variables X and Y?

E[X + Y] = E[X] + E[Y].

If a random variable X is scaled by a factor a, how is the variance affected?

Var(aX) = a² · Var(X).

How do you find the expected value of a function f(X) of a random variable X?

E[f(X)] = Σ f(xi) · pX(xi).

In the context of expected value, what does E[X²] represent?

It is the expected value of the square of the random variable X.

How do you calculate the expected value of a linear transformation of a random variable?

E[aX + b] = a · E[X] + b.