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1) For a continuous random variable \(X\), \(P(X=a)=0\) for any single number \(a\).
2) The CDF \(F(x)\) never decreases and satisfies \(\lim_{x\to-\infty}F(x)=0\) and \(\lim_{x\to+\infty}F(x)=1\).
3) The value of a PDF \(f(x)\) is always between 0 and 1.
4) The total area under any valid PDF is 1: \(\int_{-\infty}^{\infty} f(x)\,dx=1\).
5) For continuous \(X\), \(P(a\lt X\le b)=F(a)-F(b)\).
6) If the CDF \(F\) is differentiable, then the PDF is \(f(x)=F'(x)\).
7) The mean (expected value) of \(X\) is \(E[X]=\int_{-\infty}^{\infty} x\,f(x)\,dx\) when the integral exists.
8) \(\operatorname{Var}(X)=E[X]^2 - E[X^2]\).
9) The standard deviation is the square of the variance.
10) A median \(m\) satisfies \(F(m)=0.5\) for a continuous distribution.
11) If \(X\sim \text{Uniform}(a,b)\), then \(E[X]=(a+b)/2\) and \(\operatorname{Var}(X)=(b-a)^2/12\).
12) If \(Y=cX\) with constant \(c>0\), then \(\operatorname{Var}(Y)=c\,\operatorname{Var}(X)\).
13) The probability up to \(x\) equals the area under the CDF curve from \(-\infty\) to \(x\).
14) For an Exponential(\(\lambda\)) distribution (rate per unit time), \(E[T]=1/\lambda\) and \(F(t)=1-e^{-\lambda t}\) for \(t\ge0\).
15) If one PDF has a larger height than another at a particular \(x\), then its CDF must also be larger at that same \(x\).