GATE2025_CS1_Q58

Consider a probability distribution given by the density function P(x).

P(x) = { cx², for 1 ≤ x ≤ 4
0,     for x < 1 or x > 4

The probability that x lies between 2 and 3, i.e., P(2 ≤ x ≤ 3) is ______. (rounded off to three decimal places)

✅ Final Answer:

The correct answer is 0.302.

🔹 Short Answer:

First, find the constant 'c' by ensuring the total probability is 1. The integral of cx² from 1 to 4 is 1, which gives c = 1/21. Then, calculate the required probability by integrating (1/21)x² from 2 to 3. This results in (1/63) * [3³ - 2³] = 19/63. As a decimal, this is approximately 0.30158, which rounds to 0.302.

🔸 Long Answer:

This problem involves a continuous probability distribution and requires two main steps: first, normalizing the probability density function (PDF) to find the constant 'c', and second, integrating the normalized PDF over the specified interval to find the probability.

Step 1: Normalize the Probability Density Function (PDF)

A fundamental property of any PDF is that the total area under its curve must be equal to 1. This means the integral of P(x) over its entire domain must be 1.

∫_-∞^∞ P(x) dx = 1

Since P(x) is only non-zero between x=1 and x=4, our integral simplifies:

∫₁⁴ cx² dx = 1

Now, we solve this integral for 'c':

c [x³/3]₁⁴ = 1
c (4³/3 - 1³/3) = 1
c (64/3 - 1/3) = 1
c (63/3) = 1
21c = 1 ⟹ c = 1/21

So, the complete PDF is P(x) = (1/21)x² for 1 ≤ x ≤ 4.

Step 2: Calculate the Required Probability P(2 ≤ x ≤ 3)

The probability that the random variable 'x' falls within a certain range is the area under the PDF curve over that interval. We find this by integrating the normalized PDF from 2 to 3.

P(2 ≤ x ≤ 3) = ∫₂³ (1/21)x² dx

Let's compute the definite integral:

(1/21) [x³/3]₂³
= (1/21) * (3³/3 - 2³/3)
= (1/21) * (27/3 - 8/3)
= (1/21) * (19/3)
= 19/63

Step 3: Convert to Decimal and Round

The final step is to convert the fraction to a decimal and round it to three decimal places as requested.

19 ÷ 63 ≈ 0.301587...

Rounding to three decimal places, we look at the fourth decimal digit (5). Since it is 5 or greater, we round up the third digit.

0.302

This falls within the official GATE answer range of 0.300 to 0.302.

Bibliography:
- Ross, S. (2014). A First Course in Probability. Pearson. (Chapter 5: Continuous Random Variables).
- Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability & Statistics for Engineers & Scientists. Pearson.

Ask your doubts in comment form our GURUJEE will reply ASAP, click notify me button so that you get notified on reply.