Question

If x is a continuous random variable then $P(x \geqslant a) =$
A.$P(x < a)$
B.$1 - P(x > a)$
C.$P(x > a)$
D.$1 - P(x \leqslant a - 1)$

Answer 1

If we are given with continuous random variables such as X, we can only calculate the probability that X lies within a certain interval; like $P(X \leqslant k)$or $P(a \leqslant X \leqslant b)$
If the probability of X to be equal to a specific value then we can’t calculate the probability i.e.
$\Rightarrow P(X = k) = 0$
This fact remains true. As, the total number of possible values for a continuous random variable X is infinite so the possibility of any one single outcome tends towards 0.

Complete step-by-step answer:
A random variable where the data can have infinite values then it is called continuous random variable. For example, the time taken for something to be done is measured by a random variable and is continuous since there are an infinite number of possible times that can be taken.
Here in this question we have given that x is continuous random variable. Now we have given that x is a continuous random variable and we have to find the value of $P(x \geqslant a)$.
let us consider that there are n number of points in total since x is continuous, therefore
the total number of possible values is infinite i.e. $n \to \infty$and hence the possibility of any single outcome i.e. a tends towards zero.
$\Rightarrow P(x = a) = \dfrac{1}{n}$
As, $n \to \infty$, therefore $\dfrac{1}{n} \to 0$. Hence,

$\Rightarrow P(x = a) = 0$ ………..(1)

Now, we have to find the value of $P(x \geqslant a)$as below:

$\Rightarrow P(x \geqslant a) = P(x > a) + P(x = a)$
Put the value of equation 1 in this we get,
$\Rightarrow P(x \geqslant a) = P(x > a) + 0$
$\Rightarrow P(x \geqslant a) = P(x > a)$
Therefore, option C is the correct option.

Note: In this question students can make the mistake of taking the possibility of a single outcome as 1 and your answer will get wrong. As probability is the favourable outcome out of the total number of outcomes so here the outcome is 1 out of n points i.e. getting a when 1 is divided by infinity, we get 0.
If we want to calculate probability of continuous random variables i.e. probability the X lies between a certain range then we need two functions:
1.The probability density function (PDF)
2.The cumulative density function (CDF)