9709 · 6.3
Continuous random variables flashcards
Revision flashcards for Cambridge 9709 Continuous random variables (syllabus 6.3). Flip, recall, then mark a real past-paper question.
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What is a continuous random variable?
A random variable that can take any value within a given range. Examples include height, weight, or time.
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What are the two key properties of a probability density function (PDF), f(x)?
1. f(x) ≥ 0 for all x. (Probability density cannot be negative). 2. The total area under the curve is 1: $\int_{-\infty}^{\infty} f(x) dx = 1$.
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For a continuous random variable X, what is the value of P(X = c) for any constant c?
P(X = c) = 0. The probability of X taking any single exact value is zero. This is because the area under the PDF at a single point is zero.
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What is the consequence of P(X = c) = 0 for inequalities?
P(a < X < b) = P(a ≤ X < b) = P(a < X ≤ b) = P(a ≤ X ≤ b). The inclusion or exclusion of endpoints does not change the probability.
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How is the probability P(a < X < b) calculated from the PDF, f(x)?
By integrating the PDF between the limits a and b: $P(a < X < b) = \int_{a}^{b} f(x) dx$.
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What is the Cumulative Distribution Function (CDF), F(x)?
The CDF gives the cumulative probability that the variable X takes a value less than or equal to x. $F(x) = P(X \le x) = \int_{-\infty}^{x} f(t) dt$.
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How can you use the CDF, F(x), to find P(a < X < b)?
$P(a < X < b) = F(b) - F(a)$. This is often quicker than integrating the PDF from scratch.
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What is the formula for the expectation (mean) of a continuous random variable X?
$E(X) = \mu = \int_{-\infty}^{\infty} x f(x) dx$. Remember to integrate over the defined range of f(x).
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What is the formula for the variance of a continuous random variable X?
$Var(X) = \sigma^2 = \int_{-\infty}^{\infty} x^2 f(x) dx - [E(X)]^2$. This is $E(X^2) - [E(X)]^2$.
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What is a common mistake when calculating Var(X)?
Forgetting to square the value of E(X) at the end. The formula is $E(X^2) - [E(X)]^2$, not $E(X^2) - E(X)$.
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What defines a uniform (or rectangular) distribution for a variable X over the interval [a, b]?
The PDF is constant for a ≤ x ≤ b and zero elsewhere. $f(x) = \frac{1}{b-a}$ for $x \in [a, b]$.