Q.56 A random variable X and its probability distribution is given below. The value of
P(X<5) is _______. (rounded off to one decimal place)
SEO-Friendly Introduction
A common exam problem in probability asks you to use the probability distribution of a random variable X and find P(X<5) from a table of values. This type of question checks your understanding of discrete distributions, the requirement that total probability equals 1, and how to combine event probabilities correctly. By solving for the constant k and summing the appropriate probabilities, you can quickly obtain the correct value of P(X<5) and avoid typical mistakes in competitive exams.
Question Details
The table gives the probability mass function of a discrete random variable X:
| X | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|---|
| P(X) | 0 | k | 2k | 2k | 3k | k2 | 2k2 | 7k2 + k |
The task is to find P(X<5), which means the probability that X takes any value less than 5, i.e., X=0,1,2,3,4.
Step-by-Step Solution
Step 1: Find the value of k
Because this is a valid probability distribution, the sum of all probabilities must be 1.
P(X=0) + P(1) + P(2) + P(3) + P(4) + P(5) + P(6) + P(7) = 1
Substitute the given probabilities:
0 + k + 2k + 2k + 3k + k² + 2k² + 7k² + k = 1
Combine like terms:
- Linear in k: k + 2k + 2k + 3k + k = 9k
- Quadratic in k: k² + 2k² + 7k² = 10k²
10k² + 9k – 1 = 0
Solve the quadratic equation using the formula k = [-b ± √(b² – 4ac)] / (2a) with a=10, b=9, c=-1:
k = [-9 ± √(81 + 40)] / 20 = [-9 ± √121] / 20 = [-9 ± 11] / 20
This gives two solutions:
- k = (2)/20 = 0.1
- k = (-20)/20 = -1
Since a probability constant must be non-negative, the only acceptable value is k = 0.1.
Step 2: Compute P(X < 5)
P(X<5) = P(0) + P(1) + P(2) + P(3) + P(4)
P(X<5) = 0 + k + 2k + 2k + 3k = 8k
Insert k=0.1:
P(X<5) = 8(0.1) = 0.8
Rounded to one decimal place, the answer is P(X<5) = 0.8.
Exam Tips for CSIR NET
- Always verify the probability distribution sums to 1 before computing specific probabilities.
- Solve for constants like k using the normalization condition first.
- P(X<5) only includes values strictly less than 5 (X=0 to 4).
- Discard negative solutions for probability parameters.
- Double-check quadratic equation solutions using the discriminant.
