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

Step 1: Recall the standard form of each named distribution.
Binomial distribution gives the probability of exactly x successes in n independent trials, each with success probability p: \[ f(x) = \binom{n}{x}p^x(1-p)^{n-x}, \quad x=0,1,2,...,n \]
Poisson distribution models the count of rare events in a fixed interval, with mean rate \( \mu \): \[ f(x) = \frac{\mu^x}{x!}\exp(-\mu), \quad x=0,1,2,... \]
Normal distribution is the bell curve for a continuous variable with mean \( \mu \) and standard deviation \( \sigma \): \[ f(x) = \frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right), \quad \sigma>0 \]
Exponential distribution models the waiting time between events at a constant rate \( \lambda \): \[ f(x) = \lambda\exp(-\lambda x), \quad x>0 \]

Step 2: Match P, Binomial, to its formula.
Formula (2) has the binomial coefficient \( \binom{n}{x} \) with \( p^x(1-p)^{n-x} \), and x runs over the discrete set 0,1,2,...,n. This is the binomial form, so P → 2.

Step 3: Match Q, Poisson, to its formula.
Formula (3) has \( \mu^x/x! \) with \( \exp(-\mu) \), the classic Poisson form, with x running over all non-negative integers with no upper limit n. So Q → 3.

Step 4: Match R, Normal, to its formula.
Formula (1) has \( \exp\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right) \) divided by \( \sigma\sqrt{2\pi} \), the bell curve density with continuous x. So R → 1.

Step 5: Match S, Exponential, to its formula.
Formula (4), \( \lambda\exp(-\lambda x) \) for \( x>0 \), is the standard exponential density with a single rate parameter and no factorial or binomial term. So S → 4.

Step 6: Check the wrong options.
Option B sends P to formula 1 (Normal) and R to formula 2 (Binomial), a swap.
Option C sends P to formula 3 (Poisson) and Q to formula 2 (Binomial), also a swap.
Option D sends Q to formula 4 (Exponential) and S to formula 1 (Normal), mixing a discrete and a continuous distribution.

Final Answer:
The correct matching is P → 2, Q → 3, R → 1, S → 4.
\[ \boxed{P \to 2,\ Q \to 3,\ R \to 1,\ S \to 4} \]