Continuous Random Variable & Probability Density Function | Statistics and Probability | By GP Sir

Summary

This video covers key concepts in probability and integration, focusing on calculating the likelihood of events such as tubes being replaced or not within a given time frame. The lecturer walks through the steps of solving conditional probability problems, using integration techniques to find the probability of certain outcomes. Practical examples from competitive exams like CSIR NET, IIT JAM, and GATE are included, along with a focus on using resources like the MathScare app for exam preparation. The video is aimed at helping students master these concepts with clear explanations and detailed solutions.

Takeaways

• 😀 The problem involves calculating the probability of tubes being replaced or not within a given time frame.
• 😀 The probability of one tube not being replaced within 150 hours is calculated using integration and is found to be 1/3.
• 😀 For three tubes, the probability that none of them are replaced is (1/3)^3 = 1/27.
• 😀 The probability of all three tubes being replaced is (2/3)^3 = 8/27.
• 😀 Conditional probability is discussed using the exponential distribution function e^(-x).
• 😀 The formula for conditional probability is P(X > 2 | X > 1) = P(X > 2 and X > 1) / P(X > 1).
• 😀 For the conditional probability, the intersection of events X > 2 and X > 1 is equivalent to just X > 2.
• 😀 The calculation involves integration of the function e^(-x) over different intervals to find the probabilities.
• 😀 After solving the integration, the conditional probability P(X > 2 | X > 1) is found to be 1/e.
• 😀 The video also provides insights into the steps and methodology for solving similar probability problems.
• 😀 Additional learning resources are available through the MathScare app, including playlists for statistics and probability, along with Telegram updates for courses.

Q & A

What is the probability that none of the three tubes are replaced within the first 150 hours?

The probability that none of the three tubes are replaced is calculated by multiplying the probability for each tube not being replaced. Since the probability for one tube is 1/3, the probability for all three tubes is (1/3) × (1/3) × (1/3) = 1/27.

How do you calculate the probability that all three tubes are replaced within the first 150 hours?

To calculate the probability that all three tubes are replaced, first find the probability of one tube being replaced, which is 1 - 1/3 = 2/3. Then, multiply this probability for all three tubes: (2/3) × (2/3) × (2/3) = 8/27.

What is the formula used to calculate conditional probability in the video?

The formula for conditional probability used in the video is P(A | B) = P(A ∩ B) / P(B), where A is the event of interest and B is the condition. In the case of the video, it is the probability that X > 2 given that X > 1.

What are the integration limits used to solve for the conditional probability that X > 2 given X > 1?

The integration limits for solving the conditional probability are from 2 to infinity for P(X > 2) and from 1 to infinity for P(X > 1), as shown in the integration of the exponential function e^(-x).

What is the value of e^(-∞) in the calculations provided in the video?

In the calculations, e^(-∞) is considered to be 0, which simplifies the integral and leads to the final result.

What is the significance of e^(-x) in the context of the video?

e^(-x) is the probability density function of an exponential distribution, which is used to model the time between events in a Poisson process, such as the time until a tube is replaced.

How do you simplify the expression 1/e^2 from the conditional probability calculation?

After integrating and simplifying, you get the expression 1/e^2. Taking the reciprocal of this gives 1/e, which is the final value of the conditional probability.

What is the probability of a tube being replaced within the first 150 hours if the probability of not being replaced is 1/3?

If the probability of a tube not being replaced is 1/3, then the probability of a tube being replaced within 150 hours is 1 - 1/3 = 2/3.

What method is used to calculate the probability for multiple tubes being replaced in the video?

The method used involves multiplying the individual probabilities of replacement for each tube. Since the events are independent, the probability of all three tubes being replaced is the product of the individual probabilities.

How does the concept of conditional probability apply to this problem in the video?

The concept of conditional probability is applied by finding the probability that X > 2 given that X > 1, which involves calculating the intersection of the two events and dividing by the probability of the condition (X > 1).