Question Type: Probability and Statistics
This question asks to identify the probability density function (PDF) associated with discrete random variables. A probability density function describes the likelihood of a random variable taking on a given value. However, the term "density" is more accurately applied to *continuous* random variables; for discrete variables, we talk about the probability *mass* function (PMF).
Let's examine the options:
- Cumulative distribution: This function gives the probability that a random variable is less than or equal to a certain value. It applies to both discrete and continuous variables, but it's not a PDF/PMF itself.
- Gaussian distribution (Normal distribution): This is a continuous probability distribution, not suitable for discrete random variables.
- Poisson distribution: This is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known average rate and independently of the time since the last event. This is the correct answer.
- Rayleigh distribution: This is a continuous probability distribution, not suitable for discrete random variables.
Core Logic/Pattern: The core concept is understanding the difference between continuous and discrete probability distributions and their associated functions. The Poisson distribution is specifically designed for modeling discrete events.
Eliminating Incorrect Options: Options 2 and 4 are continuous distributions and are thus incorrect. Option 1 is a cumulative distribution, describing the accumulated probability up to a certain point, not the probability of a specific value, therefore it is also incorrect.
Therefore, the correct answer is the Poisson distribution.
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.
