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Random Variables, PMFs, PDFs & CDFs

Useful preparation: Conditional Sample Spaces

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Meaning & notation

A random variable turns an uncertain outcome into a number. Its distribution assigns probabilities to sets of possible numbers. For isolated outcomes we add masses; for a variable with a density we integrate area. The cumulative distribution function unifies both descriptions by asking how much probability has accumulated at or below a chosen threshold.

Probability mass function: For a discrete random variable X, $\displaystyle p\, \left(x\right) = P\, \left(X = x\right)$ . Its values are nonnegative and sum to one over the possible values.
Probability density function: A density f is nonnegative and integrates to one. Integrating f over an interval gives its probability; f(x) itself is not the probability of one point.
Cumulative distribution function: $\displaystyle F\, \left(x\right) = P$ (X at most x) is nondecreasing and right-continuous, approaching zero and one at the two infinite ends. It exists for every real-valued random variable.

Conditions before calculation

A proposed mass or density must be nonnegative everywhere on its support and have total probability one. State the support and give zero mass or density outside it.
A binomial count requires a fixed number n of independent trials, each with the same success probability p and two possible classifications. Sampling without replacement generally violates independence.
For a variable with a density, each single point has probability zero. For any distribution, P( $\displaystyle a < X$ at most b)= $\displaystyle F\, \left(b\right) - F\, \left(a\right)$ ; changing endpoint inclusion can matter for discrete distributions.

Why the method works

1. Choose the correct model

Countable outcome values suggest a mass function. Measurements modeled continuously may use a density. Mixed distributions can have both jumps and continuous portions, so do not force every distribution into one pure type.

2. Normalize before calculating

The total area or sum must equal one. A density may exceed one if its support is narrow enough; only integrated probabilities must remain between zero and one.

3. Use the CDF to subtract sets

Removing outcomes at or below a from outcomes at or below b leaves precisely the half-open interval.

P(a<X\le b)=F(b)-F(a)

Worked examples

EXAMPLE 1

Three independent sensors each signal with probability 0.2. Find the probability that exactly one signals.

Let X count signals. The fixed trial count, independence, and common probability give a binomial model.
There are three choices for the successful sensor. Each specific pattern has the same probability.
$P(X=1)=\binom31(0.2)(0.8)^2$
Multiply to get the mass; the four masses for counts zero through three also sum to one.
$P(X=1)=0.384,\quad 0.512+0.384+0.096+0.008=1$

Conclusion: The probability is 0.384, provided the independence assumption is appropriate.

EXAMPLE 2

Let $\displaystyle f\, \left(x\right) = c\, x$ on [0,2] and zero elsewhere. Find c and P( $\displaystyle 1 < X$ at most 2).

Normalize the density by integrating over its entire support.
$1=\int_0^2 cx\,dx=2c,\qquad c=\frac12$
Integrate from zero to a threshold inside the support to get the CDF there.
$F(x)=\frac{x^2}{4}\quad(0\le x\le2)$
Subtract the CDF values; the endpoints carry no mass.
$P(1<X\le2)=F(2)-F(1)=1-\frac14=\frac34$

Conclusion: The probability is $\displaystyle \frac{3}{4}$ . Complete the CDF with zero for $\displaystyle x < 0$ and one for $\displaystyle x > 2$ .

Pitfalls & counterexamples

Tempting claim: A probability density cannot exceed one.

Correction: Density is probability per unit length, so only its area is bounded by one.

Test the claim: The uniform density on [0,0.2] equals 5 but has total area 5 times $\displaystyle 0.2 = 1$ .

Check your understanding

If X has a density, what is $\displaystyle P\, \left(X = 1.5\right)$ ?

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Compare your reasoning

Zero.

A single point has zero interval length; a positive density value does not create a point mass.

Connect this idea

Distributions supply the weights for expectation and variance. Their CDFs support quantiles, simulation, and probability calculations used in statistical inference.

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Reference reading

Use these texts to study the underlying definitions and theorems in greater depth. The explanations and examples above are written for Numerive.