Match the distribution with its corresponding probability density/mass function.
Distribution type Probability density/mass function (P) Binomial Distribution (1) $f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right), \sigma > 0$ (Q) Poisson Distribution (2) $f(x) = \binom{n}{x} p^x (1-p)^{n-x}, \quad x = 0, 1, 2, \dots, n$ (R) Normal Distribution (3) $f(x) = \frac{\mu^x}{x!} \exp(-\mu), \quad x = 0, 1, 2, \dots$ (S) Exponential Distribution (4) $f(x) = \lambda \exp(-\lambda x), \quad x > 0$
Distribution PDF/PMF Identification
This solution identifies and matches each given probability distribution with its correct probability density function (PDF) or probability mass function (PMF).
Function Analysis for Distributions
- Binomial Distribution (P): This discrete distribution models the number of successes in a fixed number of independent trials ($n$) with probability of success ($p$). Its PMF is $f(x) = \binom{n}{x} p^x (1-p)^{n-x}$ for $x = 0, 1, \dots, n$. This aligns with function (2).
- Poisson Distribution (Q): This discrete distribution counts events in a fixed interval at a known average rate ($\mu$). Its PMF is $f(x) = \frac{\mu^x}{x!} \exp(-\mu)$ for $x = 0, 1, 2, \dots$. This matches function (3).
- Normal Distribution (R): A key continuous distribution known for its symmetric, bell-shaped curve. Its PDF is $f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right)$ for $\sigma > 0$. This corresponds to function (1).
- Exponential Distribution (S): This continuous distribution models the time until an event occurs in a Poisson process. Its PDF is $f(x) = \lambda \exp(-\lambda x)$ for $x > 0$. This matches function (4).
Matching Distributions and Functions
| Distribution Type | Probability Function |
| (P) Binomial Distribution | (2) $f(x) = \binom{n}{x} p^x (1-p)^{n-x}, \quad x = 0, 1, 2, \dots, n$ |
| (Q) Poisson Distribution | (3) $f(x) = \frac{\mu^x}{x!} \exp(-\mu), \quad x = 0, 1, 2, \dots$ |
| (R) Normal Distribution | (1) $f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right), \sigma > 0$ |
| (S) Exponential Distribution | (4) $f(x) = \lambda \exp(-\lambda x), \quad x > 0$ |
Final Match
The correct pairings are: Binomial (P) with function (2), Poisson (Q) with function (3), Normal (R) with function (1), and Exponential (S) with function (4).
This corresponds to the selection: P $\rightarrow$ 2, Q $\rightarrow$ 3, R $\rightarrow$ 1, S $\rightarrow$ 4.
Important Questions from Discrete Distributions
The value of a and b so that the following is probability mass function
X: 0 1 2 P(X = x): 3a 3b 4b with mean 1.1, is:
Digital data received from a sensor can fill up 0 to 32 buffers. Let the sample space be
S = {0, 1, 2, .........., 32} where the sample j denote that j of the buffers are full and \(p\left( i \right) = \frac{1}{{561}}\left( {33 - i} \right)\)
. Let A denote the event that the even number of buffers are full. Then p(A) is :If X is a Poisson random variate with mean 3, then P(|X- 3| < 1) will be:
Let x ∼ N(μ, σ2) If μ2 = σ2, (μ > 0), then the value of P(X < -μ | X < μ) in terms of cumulative function N (0, 1) is:
Consider a binomial random variable X. If X1, X2,...Xn are independent and identically distributed samples from the distribution of X with sum \(Y = \mathop \sum \limits_{i = 1}^n {X_i}\) then the distribution of Y as n → ∞ can be approximated as.
