Question

Match the distribution with its corresponding probability density/mass function.

Distribution type	Probability density/mass function
(P) Binomial Distribution	(1) \( f(x) = \dfrac{1}{\sigma\sqrt{2\pi}}\exp\left(-\dfrac{1}{2}\left(\dfrac{x-\mu}{\sigma}\right)^2\right), \ \sigma>0 \)
(Q) Poisson Distribution	(2) \( f(x) = \dbinom{n}{x}p^x(1-p)^{n-x}, \ x=0,1,2,...,n \)
(R) Normal Distribution	(3) \( f(x) = \dfrac{\mu^x}{x!}\exp(-\mu), \ x=0,1,2,... \)
(S) Exponential Distribution	(4) \( f(x) = \lambda\exp(-\lambda x), \ x>0 \)

• A. P → 2, Q → 3, R → 1, S → 4
• B. P → 1, Q → 3, R → 2, S → 4
• C. P → 3, Q → 2, R → 1, S → 4
• D. P → 2, Q → 4, R → 3, S → 1

Hint:

Split the four formulas into discrete (pmf) and continuous (pdf) types, then match by the presence of a binomial coefficient, factorial, squared exponential, or plain exponential term.

Answer 1

The Correct Option is A

A different way to solve this matching question is to first split the four functions into discrete (probability mass functions, for countable outcomes) and continuous (probability density functions, for outcomes that can be any real value), then match names within each group.

Sorting into discrete versus continuous: formula (2), $f(x)=\binom{n}{x}p^x(1-p)^{n-x}$ for $x=0,1,2,...,n$, and formula (3), $f(x)=\frac{\mu^x}{x!}\exp(-\mu)$ for $x=0,1,2,...$, both restrict x to whole numbers, so both are discrete. Formula (1), the bell shaped exponential of a square, and formula (4), $\lambda\exp(-\lambda x)$ for $x>0$, both allow x to be any real value in their range, so both are continuous.

Among the distribution types, Binomial (P) and Poisson (Q) count discrete events, so they must map to formulas (2) and (3). Normal (R) and Exponential (S) describe continuous quantities, so they must map to formulas (1) and (4).

1. Binomial (P): needs a fixed number of trials n and a binomial coefficient, which only formula (2) has, so P maps to 2.
2. Poisson (Q): needs a factorial in the denominator and no upper limit n, which is formula (3), so Q maps to 3.
3. Normal (R): needs the squared deviation from the mean divided by the standard deviation inside the exponential, along with the $\sigma\sqrt{2\pi}$ normalizing term, which is formula (1), so R maps to 1.
4. Exponential (S): needs a single decaying exponential in x with rate $\lambda$ and no square or factorial, which is formula (4), so S maps to 4.

This gives the same pairing found by sorting on formula shape: P to 2, Q to 3, R to 1, S to 4.

Let's summarize:

• Formulas restricted to whole numbers (2 and 3) belong to the discrete distributions, Binomial and Poisson.
• Formulas free to take any real value (1 and 4) belong to the continuous distributions, Normal and Exponential.
• Within each pair, the binomial coefficient identifies formula (2) as Binomial and the $\mu^x/x!$ term identifies formula (3) as Poisson; the squared exponential identifies formula (1) as Normal and the plain exponential identifies formula (4) as Exponential.

So the matching P to 2, Q to 3, R to 1, S to 4 is correct, which is option (A).