Question

A continuous random variable X has p.d.f. \({\rm{f(x)}}\) ,then:
(a) \({\rm{0 }} \le {\rm{ f(x) }} \le {\rm{ 1}}\)
(b) \({\rm{f(x) }} \ge {\rm{ 0}}\)
(c) \({\rm{f(x) }} \le {\rm{ 1}}\)
(d) \({\rm{0 < f(x) < 1}}\)

Answer 1

Hint: To solve this question, we must first know what is a continuous random variable and what p.d.f and what does it stand for. We basically have to find the range of \[{\rm{f}}\left( {\rm{x}} \right)\] in the given question and then check the option one by one.

Complete step-by-step answer:
A probability distribution function is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events. Now, we should also know what a continuous random variable is. A continuous random variable is a function on the outcomes of some probabilistic experiment which takes value in a continuous set. In this question, we have to determine what can be the range of \[{\rm{f}}\left( {\rm{x}} \right)\] if it is a probability distribution function. Now we know that the probability distribution function is nothing but the distribution of probabilities of continuous random variables. So the minimum value of \[{\rm{f}}\left( {\rm{x}} \right)\] can be zero when the probability will be zero. The maximum value of \[{\rm{f}}\left( {\rm{x}} \right)\] will be one when the value of the probability will be zero. So the range of the \[{\rm{f}}\left( {\rm{x}} \right)\] will be \({\rm{0 }} \le {\rm{ f(x) }} \le {\rm{ 1}}\) . Now, we are going to check each option.
Option (a): Option (a) is correct
Option (b): Option (b) is incorrect because if the value of \[{\rm{f}}\left( {\rm{x}} \right) \ge 0\] then it is also possible that \({\rm{f(x) }} \ge {\rm{ 1}}\) , which is not the case.
 Option (c): Option(c) is incorrect because if the value of \({\rm{f(x)}} \le {\rm{1}}\) then it is also possible that \({\rm{f(x)}} \le 0\) , which is not the case.
Option (d): Option (d) is correct because it is in the range of \[{\rm{f(x)}}\]
Hence, option (a) and (d) are correct.


Note: Instead of continuous variable function, if there is discrete variable function then also the range of \[{\rm{f(x)}}\] will be same because still the probability has range \({\rm{0 }} \le {\rm{ P }} \le {\rm{ 1}}\) .