From CDF to PDF
How does accumulated probability reveal local probability density?
How this lesson unfolds.
Begin with What is the probability that a uniform point lies before x? The sequence then connects The CDF records accumulated probability and builds toward Differentiate the CDF piece by piece.
What is the probability that a uniform point lies before x?
For X uniform on [0,1], equal lengths have equal probability. The event X≤x corresponds to the length from 0 to x.
For X uniform on [0,1], equal lengths have equal probability.The event X≤x corresponds to the length from 0 to x.
- Learning goalConnect accumulated probability, local density and differentiation.
- Helpful foundationscontinuous probability · functions · basic derivatives
- Common trapA density is a local rate per unit length, not the probability of one exact continuous value.
- Next connectionThe CDF records accumulated probability
What exactly does F(x) collect?
The cumulative distribution function F(x)=P(X≤x) records all probability to the left of x. For Uniform(0,1), it is 0 before 0, x on [0,1], and 1 after 1.
The cumulative distribution function F(x)=P(X≤x) records all probability to the left of x.
- Learning goalConnect accumulated probability, local density and differentiation.
- Helpful foundationscontinuous probability · functions · basic derivatives
- Common trapA density is a local rate per unit length, not the probability of one exact continuous value.
- Next connectionA tiny interval exposes a local rate
How much probability enters the CDF across a tiny interval?
Across [x,x+Δx], the CDF increase ΔF equals the probability in that interval. Dividing ΔF by Δx gives probability per unit length.
Across [x,x+Δx], the CDF increase ΔF equals the probability in that interval.
- Learning goalConnect accumulated probability, local density and differentiation.
- Helpful foundationscontinuous probability · functions · basic derivatives
- Common trapA density is a local rate per unit length, not the probability of one exact continuous value.
- Next connectionCDF and PDF answer different questions
What changes when we move from accumulated probability to density?
The CDF asks how much probability has accumulated by x. The PDF asks how rapidly probability is accumulating near x; density is not point probability.
- Learning goalConnect accumulated probability, local density and differentiation.
- Helpful foundationscontinuous probability · functions · basic derivatives
- Common trapA density is a local rate per unit length, not the probability of one exact continuous value.
- Next connectionCDF slope becomes PDF height
Why does a linear CDF produce a flat density?
Inside (0,1), equal moves in x create equal rises in F, so the CDF slope is 1. The PDF therefore has constant height 1.
Inside (0,1), equal moves in x create equal rises in F, so the CDF slope is 1.
- Learning goalConnect accumulated probability, local density and differentiation.
- Helpful foundationscontinuous probability · functions · basic derivatives
- Common trapA density is a local rate per unit length, not the probability of one exact continuous value.
- Next connectionDifferentiate accumulated probability
Where does the PDF come from mathematically?
Where the CDF is differentiable, the density is its derivative: f(x)=F′(x). Accumulation and local rate are two views of the same distribution.
Where the CDF is differentiable, the density is its derivative: f(x)=F′(x).
- Learning goalConnect accumulated probability, local density and differentiation.
- Helpful foundationscontinuous probability · functions · basic derivatives
- Common trapA density is a local rate per unit length, not the probability of one exact continuous value.
- Next connectionDifferentiate the CDF piece by piece
What PDF does the Uniform(0,1) CDF produce?
The CDF is flat outside [0,1] and has slope 1 inside, so f(x)=1 on (0,1) and 0 outside. Endpoint values do not change any continuous probability.
The CDF is flat outside [0,1] and has slope 1 inside, so f(x)=1 on (0,1) and 0 outside.
- Learning goalConnect accumulated probability, local density and differentiation.
- Helpful foundationscontinuous probability · functions · basic derivatives
- Common trapA density is a local rate per unit length, not the probability of one exact continuous value.
- Next connectionUse the CDF to compute interval probabilities