Let X be a random variable with cumulative distribution function given by F(x)={0, if x<0 x+13, if 0x<1 1, if x1 Then the value of P(13<X<34)+P(X=0) is equal to

This question was previously asked in
CSIR-UGC (NET) Mathematical Science: Held on (2024 June)
View all CSIR NET Papers >
  1. 736
  2. 1136
  3. 1336
  4. 1736

Answer (Detailed Solution Below)

Option 4 : 1736
Free
Seating Arrangement
3.4 K Users
10 Questions 20 Marks 15 Mins

Detailed Solution

Download Solution PDF

Concepts Used:

1. Cumulative Distribution Function (CDF):

The CDF F(x) gives the probability that the random variable X takes a value less than or equal to x . That is, F(x) = P(X ≤ x) .

2. Finding Probability Using the CDF:

The probability that the random variable X lies within a certain interval (a, b] is given by:
     
 P(a < X ≤ b) = F(b) - F(a)

3. Probability at a Specific Point (Jump Discontinuity):

The probability at a specific point x = c is the difference in the CDF just to the right and just to the left of c :
     
P(X = c) = F(c+) - F(c-)

Explanation -

We are given a cumulative distribution function (CDF) F(x) of a random variable X as:

F(x) = {0,if x<0x+13,if 0x<11,if x1

From the definition of the probability from the CDF:  P(a < X  b) = F(b) - F(a)

In this case, we need to calculate F(34)and F(13).

Since 013<1  and 034<1 , we use the formula F(x)=x+13 for both 1/3  and 3/4 :

F(13)=13+13=433=49

F(34)=34+13=743=712

Thus, the probability P(13<X34) is:

P(13<X34)=F(34)F(13)=71249

= 21361636=536

The probability at a point is the jump in the CDF at that point. We need to calculate F(0+) - F(0-) .

From the CDF definition: F(0+) = F(0) =0+13=13

⇒ F(0-) = 0

Thus, P(X=0)=F(0+)F(0)=130=13=1236

 

Now, we add the two results:

P(13<X34)+P(X=0)=536+1236=1736

Thus, the final answer is 17/36.

Latest CSIR NET Updates

Last updated on Jun 5, 2025

-> The NTA has released the CSIR NET 2025 Notification for the June session.

-> The CSIR NET Application Form 2025 can be submitted online by 23rd June 2025

-> The CSIR UGC NET is conducted in five subjects -Chemical Sciences, Earth Sciences, Life Sciences, Mathematical Sciences, and Physical Sciences. 

-> Postgraduates in the relevant streams can apply for this exam.

-> Candidates must download and practice questions from the CSIR NET Previous year papers. Attempting the CSIR NET mock tests are also very helpful in preparation.

More Statistics & Exploratory Data Analysis Questions

Get Free Access Now
Hot Links: teen patti circle teen patti real cash apk teen patti stars