4.
The probability distribution function P(x) of a continuous random variable x is shown in
the left figure. Which of the curves in the right figure best represents the cumulative
distribution function C(x) corresponding to P(x)?
a. A
b. B
c. C
d. D
How to Match a Probability Distribution Function to Its Cumulative Distribution Function in MCQs
Final Answer: The correct option is A. For the given bimodal probability density function P(x), the corresponding cumulative distribution function C(x) must be a smooth, always-increasing S-shaped curve that rises in two stages, and curve A matches this behavior.
Understanding the Question
The left figure shows a continuous bimodal probability density function (PDF) P(x) with one tall peak on the left (negative x) and a smaller peak on the right (positive x).
A cumulative distribution function (CDF) C(x) is defined as
C(x) = P(X ≤ x) = ∫−∞x P(t) dt, so it is the area under the PDF curve from left to right.
From standard properties of CDFs:
- C(x) starts near 0 for very negative x and approaches 1 for very large x.
- It is non-decreasing and continuous for a continuous random variable.
- Where the PDF is high, the CDF has a steep slope; where the PDF is low, the CDF becomes flatter.
Because the left peak of P(x) is taller and wider than the right one, more probability mass lies on the left, so the CDF should rise quickly in the negative region, then slowly between the peaks, and then rise again in the positive region, finally flattening near 1.
Why Option A Is Correct
Curve A (solid blue) starts near 0 at the far left and increases steeply in the region of the large left peak, reflecting a large accumulation of probability there.
Between the two peaks it becomes nearly flat, because almost no additional area is being added under the PDF in that interval.
Around the second, smaller peak on the right, curve A rises again but less sharply, capturing the smaller remaining probability mass, and then smoothly approaches 1 as x goes to large positive values.
This two-stage S-shape, with an early rapid rise and a later gentler rise, is exactly what is expected from the bimodal PDF sketched in the question, so A best represents the true cumulative distribution function C(x).
Why Option B Is Not Correct
Curve B (dashed yellow) stays almost flat near zero for all negative x and begins to rise only around the origin, as if almost all probability mass were located to the right of 0.
That behavior corresponds to a PDF concentrated in the positive region, which contradicts the given PDF where there is a large peak at negative x.
The left peak would already contribute significant probability before x = 0, so the CDF must have risen substantially by then, which is not shown by curve B.
Therefore, option B does not match the given probability distribution function.
Why Option C Is Not Correct
Curve C (dotted green) increases slowly across negative x and then has its major steep rise only in the interval around the right-hand peak.
This pattern would imply that the larger share of probability is located near positive x, whereas the actual PDF clearly has its taller, dominant peak on the left.
For the given PDF, the CDF should accumulate more probability earlier (in the negative region), not mainly on the right, so option C cannot be the correct cumulative distribution function.
Why Option D Is Not Correct
Curve D (dash-dot red) remains essentially at zero until just before the right-hand region and then jumps steeply to near 1 within a narrow band of positive x.
Such a CDF would belong to a PDF with almost all of its mass concentrated in that narrow positive interval, with negligible probability anywhere else.
The provided PDF, however, spreads probability across both negative and positive regions with a prominent left peak, so the CDF must have risen well before the positive interval.
Therefore, curve D is inconsistent with the structure and symmetry of the given probability distribution function.
SEO-Optimized Introduction
In many competitive exams and statistics courses, students must master matching a probability distribution function to its cumulative distribution function.
Understanding how the shape of a PDF translates into the slope and curvature of a CDF is crucial for solving graphical MCQs efficiently and accurately.
