The normal distribution has a bell-shaped PDF.

The uniform distribution has a flat PDF.

Question

The normal distribution has a bell-shaped PDF.

The uniform distribution has a flat PDF.

BlackTom AI · Accepted Answer

The normal distribution’s probability density function (PDF) is f(x) = (1/(σ√(2π))) exp(−(x−μ)²/(2σ²)), which produces the characteristic symmetric, unimodal “bell” curve centered at μ. The uniform distribution’s PDF on an interval [a,b] is f(x) = 1/(b−a) for a ≤ x ≤ b (and 0 elsewhere), which is constant (flat) across the interval.

1. Relevant knowledge points and concepts
- PDF (probability density function): gives relative likelihood of continuous random variable values; the graph shape conveys distribution features.  
- Normal (Gaussian) distribution: continuous, determined by mean μ and standard deviation σ; symmetric about μ; exponentially decaying tails; single peak → bell shape.  
- Uniform distribution (continuous): all values in [a,b] equally likely; PDF constant over the support → flat shape.

2. Step-by-step reasoning
- Statement: “The _____ distribution has a bell-shaped PDF.” A bell-shaped PDF means symmetric around a central value, single peak, decreasing tails; this identifies the normal (Gaussian) distribution because its PDF is proportional to exp(−(x−μ)²/(2σ²)), producing the bell.  
- Statement: “The _____ distribution has a flat PDF.” A flat PDF means the density does not change with x across the support, i.e., a constant value; this matches the continuous uniform distribution where density = 1/(b−a).

3. Intermediate formula steps (showing shape origin)
- Normal: f(x) = (1/(σ√(2π))) e^{−(x−μ)²/(2σ²)}. The exponent is a negative quadratic in x, so f(x) is maximal at x = μ and decreases symmetrically as |x−μ| increases → bell.  
- Uniform: f(x) = 1/(b−a) for x in [a,b]. This is independent of x across the interval → horizontal line (flat).

4. Why the correct answers are correct
- The normal’s PDF explicitly yields the bell-shaped curve; key properties (symmetry, unimodal) match “bell-shaped.”  
- The uniform’s constant PDF exactly corresponds to “flat PDF” because every x in [a,b] has equal density.

5. Why other distributions would be incorrect
- Exponential: skewed, monotone decreasing (not bell or flat).  
- Binomial/Poisson: discrete (no continuous PDF shape like bell or flat).  
- Triangular: piecewise linear peak, not flat; only resembles bell in being unimodal but not symmetric Gaussian form.

Conclusion: therefore the first blank is normal and the second blank is uniform, matching the provided answer ["normal", "uniform"].

The normal distribution has a bell-shaped PDF. The uniform distribution has a flat PDF.Multiple dropdown selections

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