Continuous random variables (HL)

Here we do the same thing as discrete random variables but the summations change to integrals.

Contents

Probability density function

In continuous distributions, the probability is specified for an interval, rather than a specific value. The area under some probability density function f(x)f(x) over an interval is calculated using calculus.

The equivalent formulas are

∫−∞∞f(x) dx=1\int_{-\infty}^\infty f(x) \d x = 1
E(X)=μ=∫−∞∞xf(x) dx\text{E}(X) = \mu = \int_{-\infty}^\infty x f(x) \d x
Var(X)=σ2=∫−∞∞(x−μ)2f(x) dx=∫−∞∞x2f(x) dx−μ2\text{Var}(X) = \sigma^2 = \int_{-\infty}^\infty (x-\mu)^2 f(x) \d x = \int_{-\infty}^\infty x^2 f(x) \d x - \mu^2

Typically, f(x)=0f(x) = 0 is non-zero only over a finite interval, so in practice only integrate over part of the domain with nonzero f(x)f(x),

Finally, probability is determined using

P(a<X<b)=P(a≤X≤b)=∫abf(x) dx\text{P}(a < X < b) = \text{P}(a \leq X \leq b) = \int_a^b f(x) \d x

As is the case for normal distributions:

P(a≤X≤b)=P(a<X<b)\text{P}(a \leq X \leq b) = \text{P}(a < X < b)

Unknown endpoints

One of upper or lower limits of integration may be missing. For example, probability of XX between 22 and xx is 0.250.25 translates to

P(2<X<x)=∫2xf(w) dw=0.25\text{P}(2 < X < x) = \int_2^x f(w) \d w = 0.25

ww is the variable of integration. This definite integral is actually a function.

g(x)=∫2xf(w) dwg(x) = \int_2^x f(w) \d w

g(x)=0.25g(x) = 0.25 may be asked to be solved by hand or on a calculator.

This method can be used to find the median, the value below which the cumulative probability is 0.50.5.

median m:    ∫−∞mf(x) dx=0.5\text{median }m: \;\; \int_{-\infty}^m f(x) \d x = 0.5

Mode

The mode is the xx-coordinate of the global maximum of f(x)f(x).

When you think about it, it’s weird actually. All precise values of xx appear with exactly 00 probability. So the mode for continuous distributions is to indicate where you see more values, rather than the value that will appear most frequently.

Tips

  • Does it want probability of being inside or outside some interval?
  • The intersection of two events is their area(s) of overlap. Conditional probability is a ratio (fraction) of areas. See also probability formulas.
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