9231 · 4.1
Continuous random variables flashcards
Revision flashcards for Cambridge 9231 Continuous random variables (syllabus 4.1). Flip, recall, then mark a real past-paper question.
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What is a continuous random variable (CRV)?
A random variable that can take any value within a given range or interval. Examples include height, weight, and time.
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What is a Probability Density Function (PDF), denoted $f(x)$?
A function that describes the relative likelihood for a continuous random variable to take on a given value. The area under the graph of the PDF over an interval gives the probability of the variable falling in that interval.
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What are the two essential properties of any PDF, $f(x)$?
1. Non-negativity: $f(x) \ge 0$ for all $x$. 2. Total probability is 1: $\int_{-\infty}^{\infty} f(x) \, dx = 1$.
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For a CRV $X$, what is the value of $P(X=a)$?
Zero. The probability of a CRV taking a single, exact value is zero. Probability is only defined over an interval, as the area under a single point is zero.
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How do you calculate $P(a < X < b)$ for a CRV?
By integrating the PDF from $a$ to $b$: $P(a < X < b) = \int_{a}^{b} f(x) \, dx$.
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What is the Cumulative Distribution Function (CDF), denoted $F(x)$?
$F(x)$ gives the cumulative probability that the variable $X$ is less than or equal to a value $x$. It is defined as $F(x) = P(X \le x) = \int_{-\infty}^{x} f(t) \, dt$.
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How can you find the PDF, $f(x)$, if you are given the CDF, $F(x)$?
By differentiating the CDF: $f(x) = F'(x) = \frac{d}{dx}F(x)$.
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What is the formula for the expectation (mean) of a CRV, $E(X)$?
$E(X) = \mu = \int_{-\infty}^{\infty} x f(x) \, dx$. This integral is taken over the entire range where $f(x)$ is non-zero.
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What is the formula for the variance of a CRV, $Var(X)$?
$Var(X) = \sigma^2 = \int_{-\infty}^{\infty} (x - \mu)^2 f(x) \, dx$, or more commonly, $Var(X) = E(X^2) - [E(X)]^2$, where $E(X^2) = \int_{-\infty}^{\infty} x^2 f(x) \, dx$.
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For a CRV, is $P(a \le X \le b)$ different from $P(a < X < b)$?
No, they are the same. Since $P(X=a)=0$ and $P(X=b)=0$, including the endpoints does not change the value of the integral.
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What is a common trap when dealing with piecewise PDFs?
Forgetting to consider all parts of the function's domain. When finding a constant like 'k' or calculating E(X), you may need to sum integrals over different intervals.
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How do 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 if the CDF is already known.
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What are the two conditions for a function f(x) to be a valid probability density function (PDF)?
1. f(x) ≥ 0 for all x. 2. The integral of f(x) over its entire domain equals 1 (i.e., ∫ f(x) dx = 1).
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How is the cumulative distribution function, F(x), related to the probability density function, f(x)?
F(x) is the integral of f(t) from the lower bound of the domain up to x. Conversely, f(x) is the derivative of F(x), i.e., f(x) = F'(x).
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State the formula for the expectation (mean), E(X), of a continuous random variable X with PDF f(x).
E(X) = ∫ x * f(x) dx, where the integral is taken over the entire domain of X.
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What is the formula for the variance, Var(X), of a continuous random variable?
Var(X) = E(X²) - [E(X)]², where E(X²) = ∫ x² * f(x) dx.
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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. Probability for a CRV is only defined over an interval, representing the area under the PDF curve. The area of a line of zero width is zero.