Discrete random variables flashcards

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What is a discrete random variable?

A variable whose value is a numerical outcome of a random phenomenon, which can only take a countable number of values.

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What are the two conditions for a valid probability distribution of a discrete random variable X?

1. $0 \le P(X=x) \le 1$ for all values of x. 2. $\sum P(X=x) = 1$.

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What is the formula for the expectation, E(X)?

$E(X) = \sum x P(X=x)$. It represents the mean or long-run average value of X.

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What is the main formula for the variance, Var(X)?

$Var(X) = E(X^2) - [E(X)]^2$.

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How do you calculate $E(X^2)$?

$E(X^2) = \sum x^2 P(X=x)$. You square each value of x *before* multiplying by its probability.

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What is the rule for $E(aX + b)$?

$E(aX + b) = aE(X) + b$. Expectation is affected by both scaling (a) and shifting (b).

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What is the rule for $Var(aX + b)$?

$Var(aX + b) = a^2 Var(X)$. Variance is only affected by scaling (a), not shifting (b).

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Why is the 'b' term ignored in $Var(aX + b)$?

Adding a constant 'b' shifts the entire distribution but does not change its spread or variability.

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Common mistake: What is the difference between $[E(X)]^2$ and $E(X^2)$?

$[E(X)]^2$ is the square of the mean. $E(X^2)$ is the mean of the squares of the values. They are not the same unless Var(X)=0.

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What does the standard deviation of a random variable represent?

The standard deviation, $\sigma = \sqrt{Var(X)}$, is a measure of the typical amount that the values of the random variable deviate from the mean, E(X). It is in the same units as X.

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If E(X) = 3.5 for a die roll, but you can't roll a 3.5, what does it mean?

E(X) is a long-run average. If you rolled the die many times and averaged the scores, the average would get very close to 3.5. It's not necessarily a possible outcome itself.