If and
are two independent gamma distributions and
then find the distribution of .
We need to find the distribution of \( U = \frac{X}{X + Y} \) when \( X \sim \text{Gamma}(\alpha, \lambda) \) and \( Y \sim \text{Gamma}(\beta, \lambda) \), given that they are independent. ### Step 1: Understanding the Joint Distribution of \(X\) and \(Y\) The probability density function (PDF) of a Gamma-distributed random variable \( X \sim \text{Gamma}(\alpha, \lambda) \) is: \[ f_X(x) = \frac{\lambda^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\lambda x}, \quad ___ ________ ___ __________ __________ ______ ____ ____ ___ ____ _____.
___ ______ ____ ___ ___ __________ __________ ______ _________ _____ ______.
____ __________ ______ ______ ___ __________ __________ _________ ___ _______.
_______ ____ _________ ___ ____ _______ ________ _________ _______ _________.
_________ ________ ___ _____ ________ _________ __________ ________ ___ _________ _______.
___ _________ ________ _________ _____ _______ __________ _________ ______ _______ _____ _________.
___ ____ ________ _____ _______ __________ ______ _________ _________ ____.
_______ _________ ______ _____ ____ __________ ________ _______ ___ __________ ___.
_____ ____ ____ ____ ______ _____ _______ ________ ____ ______.
__________ _____ ___ ___ _________ ___.
_______ _______ _____ __________ ______.
____ _________ __________ _______ _______ __________ _____ _____ ________ _______ ___.
___ _______ _____ ______ _____.
________ ______ ___ _________ __________ _______ _________ _____.
______ _________ __________ __________ ____ ___ ___ _______.
__________ ____ ___ __________ _________ ________ ________ _____ ________ __________ _____ ___.
_____ _________ ___ ________ ________ ___ ______ ______ __________ _______ __________ ______.
____ _____ ____ _________ _______ _______ ________.
_____ _________ ________ ______ __________ ___ __________ _______ _________ _________ ____ _______.
_____ _____ ____ __________ ____ _______ _________ _______ _______ ___.
____ ________ ________ _______ _______ _____ ________ ______.
___ _______ _____ ______ ______ __________ _______ __________ ________ __________ _____.
_______ ________ _______ ____ _________ ______ _______ ___ _____ _________ ________ _____.
____ _____ _________ ____ _____.
______ __________ ____ _______ ____ ______.
_______ ________ __________ ______ ___ _____ ____ _____ _______ ____.
__________ _______ _____ ________ _______ ___ ________ ________ ______ _______ ________.
_____ _________ _____ _____ __________ _________ _________ _____ _______ __________.
_______ ________ ________ _______ _________ __________ ______ ____ _______ __________.
________ ____ ____ _________ ____ _________ _____ _________.
______ _______ _____ ____ ________.
__________ __________ __________ ___ ______ __________ ___ _________ _________ ____ _________ ___.
_____ _______ ______ __________ ________ ____ _______ __________ _____ _______ ____.
_____ __________ ___ _____ _______ ______ ___ ________.
_________ ___ ____ _______ _____ _________ ______ ____ __________ ________ _______ _____.
_________ _____ _____ ____ ______ _______ ______ ____.
_________ ____ ________ _________ ______ ___ ___.
________ ___ _____ _____ ___.
________ ______ ________ _________ ______ _______ ______.
__________.
Get Full Answer on WhatsApp
Explain the procedure of assigning probability in a continuous world of probability theory.
If and
are two independent gamma distributions and
then find the distribution of .
A hospital specializing in heart surgery. In 2022, a total of 2000 patients were admitted for treatment. The average payment made by a patient was Rs. 1,50,000 with a standard deviation of Rs. 25,000. Under the assumption that payments follow a normal distribution, answer the following questions:
i). The number of patients who paid between Rs. 1,40,000 and Rs. 1,70,000.
ii). The probability that a patient bill exceeds Rs. 1,00,000.
iii). The maximum amount paid by the lowest-paying one-third of patients.
In the study learning material (SLM), you have seen many situations where Poisson distribution is suitable and discussed some examples of such situations. Create your own example for a situation other than which are discussed in SLM. If you denote your created random variable by X then find the probability that X is less than 2.
In an election there are two candidates. Being a statistician, you are interested in predicting the result of the election. So, you plan to conduct a survey. Using the learning skill of this course answer the following question. How many people should be surveyed to be at least 90% sure that the estimate is within 0.03 of the true value?
If is a sequence of continuous random variables such that:
then prove that .
Suppose two friends Anjali and Prabhat trying to meet for a date to have lunch say between 2 pm to 3 pm. Suppose they follow the following rules for this meeting:
- Each of them will reach either on time or 10 minutes late or 20 minutes late or 30 minutes late or 40 minutes late or 50 minutes late or 1 hour late. All these arrival times are equally likely for both of them.
- Whoever of them reaches first will wait for the other to meet only for 10 minutes. If within 10 minutes the other does not reach, he/she leaves the place and they will not meet.
Find the probability of their meeting.