Reading Probabilities from CDF Graphs

Reading a CDF as a Probability Graph

A cumulative distribution function turns probability questions about a random variable \(X\) into questions about graph coordinates. The horizontal coordinate is a threshold \(t\); the vertical coordinate \(F(t)\) gives the probability that \(X\) is at most that threshold. Reading the graph correctly requires attention not only to height, but also to whether the question includes an endpoint and whether the graph has a jump there.

The key distinction is between the value of the function at \(t\) and the height approached from the left. A CDF is right-continuous, so at a jump its plotted value is the height on the upper, right-hand side. The lower height approached from the left is \(F(t-)\). These two heights answer different probability questions.

Definition: For a real-valued random variable \(X\), its cumulative distribution function is \(F(t)=P(X\le t)\). The left-hand limit at \(t\) is denoted by \(F(t-)=\lim_{x\uparrow t}F(x)\). On a graph, read \(F(t)\) at the function's actual value at \(t\), and read \(F(t-)\) as the height approached from smaller \(x\)-values.

On a graph with a jump, a filled dot at the upper endpoint marks the value \(F(t)\). The lower endpoint of the jump indicates \(F(t-)\); it may be shown as an open dot. The open or filled drawing is a visual aid, not a separate probability rule: the important point is which height is the function value and which is the left-hand limit.

Threshold Probabilities

At a threshold \(t\), the CDF directly gives \(P(X\le t)\). The probability of exceeding \(t\) is the remaining probability, while the probability of being strictly below \(t\) is given by the left-hand limit. These distinctions matter when a graph jumps at the threshold.

Threshold rules: \(P(X\le t)=F(t)\), \(P(X>t)=1-F(t)\), \(P(X<t)=F(t-)\), and \(P(X\ge t)=1-F(t-)\).

For the last two rules, the left-hand limit is the probability below \(t\), not the probability at or below \(t\). The event \(X=t\) is excluded from \(X<t\), but included in \(X\le t\). If the graph is continuous at \(t\), the two heights agree; at a jump, they do not.

Worked Example: Reading Strict and Weak Thresholds

A plotted CDF is constant at \(0\) for \(x<0\), jumps to \(0.25\) at \(x=0\), rises to \(0.60\) at \(x=2\), and then continues upward. At \(t=0\), the graph shows a lower left-hand height of \(0\) and a filled function value of \(0.25\). Find the four threshold probabilities at zero.

The function value is \(F(0)=0.25\), so \(P(X\le 0)=0.25\).

The probability of being strictly below zero is the left-hand height, \(F(0-)=0\). Thus \(P(X<0)=0\).

Taking complements gives \(P(X>0)=1-F(0)=1-0.25=0.75\), and \(P(X\ge 0)=1-F(0-)=1-0=1\).

The difference between \(P(X\ge0)\) and \(P(X>0)\) is \(0.25\), exactly the difference between the two heights at zero.

Interval Probabilities and Endpoint Inclusion

For an interval, identify its left and right endpoints first, then check whether each endpoint is included. The CDF values at the endpoints determine the probability; subtracting the wrong heights changes whether an endpoint's probability is counted.

Interval rule: For real numbers \(a<b\), \(P(a<X\le b)=F(b)-F(a)\). The other endpoint conventions follow by using \(F(a-)\) or \(F(b-)\) when an endpoint is excluded or included as appropriate.

Here are the four common forms. This table is especially useful when one or both endpoints coincide with jumps.

Event	Probability from the CDF
\(a<X\le b\)	\(F(b)-F(a)\)
\(a\le X\le b\)	\(F(b)-F(a-)\)
\(a\le X<b\)	\(F(b-)-F(a-)\)
\(a<X<b\)	\(F(b-)-F(a)\)

The reason for the basic interval rule is that \(\{X\le a\}\) is contained in \(\{X\le b\}\). Removing the first event from the second leaves exactly \(\{a<X\le b\}\). The endpoint variants use the same set subtraction, with \(X<a\) or \(X<b\) represented by the relevant left limit.

Result and derivation: Since \(\{X\le b\}\) is the disjoint union of \(\{X\le a\}\) and \(\{a<X\le b\}\), additivity of probability gives \(F(b)=F(a)+P(a<X\le b)\). Rearranging yields \(P(a<X\le b)=F(b)-F(a)\). Also, \(\{X<t\}\) is the increasing union of events \(\{X\le t-1/n\}\) over positive integers \(n\); continuity of probability for increasing events therefore gives \(F(t-)=P(X<t)\). Replacing \(F(t)\) by \(F(t-)\) at an excluded upper endpoint, or at a lower endpoint whose values below it must be removed, gives the other entries in the table.

Worked Example: An Interval Crossing a Sloping Section

Suppose a CDF graph is constant at zero for \(x<0\), rises in a straight line from \((0,0)\) to \((4,1)\), and is constant at one for \(x\ge4\). Find \(P(1<X\le3)\) and \(P(1\le X\le3)\).

Along the straight segment, the graph height at \(x\) is \(x/4\). Therefore \(F(1)=1/4\), \(F(3)=3/4\), and there are no jumps at either endpoint, so \(F(1-)=F(1)\).

For the first event, the left endpoint is excluded and the right endpoint is included: \(P(1<X\le3)=F(3)-F(1)=3/4-1/4=1/2\).

For the second event, the left endpoint is included: \(P(1\le X\le3)=F(3)-F(1-)=3/4-1/4=1/2\).

The answers agree because the graph has no jump at \(1\), so including or excluding that single endpoint adds no probability.

Reading a Point Probability from a Jump

At a jump, the vertical distance between the left-hand height and the actual function value is the probability assigned to that exact \(x\)-value. The following identity explains why; it also gives a direct way to calculate a point probability from a plotted CDF.

Result and proof: For every real \(t\), \(P(X=t)=F(t)-F(t-)\). Indeed, \(\{X<t\}\) is contained in \(\{X\le t\}\), and the difference between these events is exactly \(\{X=t\}\). Thus \(P(X\le t)-P(X<t)=P(X=t)\). Since \(F(t)=P(X\le t)\) and \(F(t-)=P(X<t)\), the identity follows. \(\square\)

This calculation is local to the threshold: read the upper height at \(t\), read the height approached from the left, and subtract lower from upper. A flat or sloping appearance elsewhere on the graph does not change this procedure.

Worked Example: Finding a Point Probability and a Nearby Interval

At \(x=2\), a CDF graph approaches height \(0.40\) from the left and has function value \(0.70\). At \(x=4\), it has value \(F(4)=0.90\), with no jump there. Find \(P(X=2)\), \(P(2<X\le4)\), and \(P(2\le X\le4)\).

The point probability is the jump size: \(P(X=2)=F(2)-F(2-)=0.70-0.40=0.30\).

The interval excluding \(2\) and including \(4\) is read using the function values: \(P(2<X\le4)=F(4)-F(2)=0.90-0.70=0.20\).

Including \(2\) adds its point probability. Equivalently, use the left-hand height at the lower endpoint: \(P(2\le X\le4)=F(4)-F(2-)=0.90-0.40=0.50\).

The check \(0.20+0.30=0.50\) confirms that the inclusive interval consists of the point \(X=2\) together with values strictly above \(2\) through \(4\).

Limits and What a Graph Window Can Tell You

A complete CDF approaches zero as the horizontal coordinate tends to negative infinity and approaches one as it tends to positive infinity. On a plotted graph, these are tail heights: far to the left the graph approaches \(0\), and far to the right it approaches \(1\). They are limits, not necessarily values at the edges of the display window.

For example, if a graph's right edge is at \(x=10\) and its height there is \(0.96\), that only tells you \(F(10)=0.96\), if the plotted point is the function value. It does not tell you that the right-tail limit is \(0.96\). A graph window may stop before the curve visibly reaches its limiting height. Use the stated tail behavior or the full graph, rather than treating the edge of the picture as infinity.

Key takeaway: At a threshold \(t\), use \(F(t)\) for events that include values through \(t\), and \(F(t-)\) for events strictly below \(t\). For interval probabilities, match the endpoint inclusions before subtracting heights; a jump is the difference \(F(t)-F(t-)\).

Common Mistakes

Using the lower jump height as \(F(t)\). The lower height is \(F(t-)\); the function value at the jump is the right-continuous, upper height.
Subtracting the same endpoint value for every interval. For example, \(P(a\le X\le b)\) uses \(F(a-)\), while \(P(a<X\le b)\) uses \(F(a)\). Check inclusion separately at each endpoint.
Assuming a jump's height is \(P(X\le t)\). The entire upper height is \(F(t)\); the point probability is only the jump size \(F(t)-F(t-)\).
Reading the graph edge as a tail limit. A displayed window is finite. The boundary limits concern \(x\) tending without bound, not the last coordinate shown.
Interpolating through a jump or a flat step. Read the plotted graph as drawn. Do not invent a sloping segment between jump endpoints or assume a CDF rises between them unless the graph shows that behavior.

Check Your Understanding

For each question, identify which graph height or heights are needed before calculating.

At \(t=3\), a CDF has \(F(3-)=0.45\) and \(F(3)=0.62\). Find \(P(X<3)\), \(P(X\le3)\), and \(P(X=3)\).
A graph gives \(F(1)=0.20\) and \(F(5)=0.85\), with no jumps at either endpoint. Find \(P(1<X\le5)\).
At \(a\), the graph jumps. Which height, \(F(a)\) or \(F(a-)\), should be subtracted to calculate \(P(a\le X\le b)\)?
A graph window ends at \(x=8\), where the plotted CDF height is \(0.93\). Does this determine \(\lim_{x\to\infty}F(x)\)? Explain.
Write the formula for \(P(a<X<b)\) in terms of \(F\) and its left-hand limits.