If is a sequence of continuous random variables such that:
then prove that .
We need to prove that \( X_n \xrightarrow{P} 0 \), which means that for every \( \epsilon > 0 \), \[ \lim_{n \to \infty} P(|X_n| \geq \epsilon) = 0. \] ### Step 1: Compute the Probability \( P(|X_n| \geq \epsilon) \) The probability _________ ____ ________ _________ ___ ______ ___ ______ ____ ____ ____.
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If and
are two independent gamma distributions and
then find the distribution of .
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.
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?
Explain the procedure of assigning probability in a continuous world of probability theory.
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.
If is a sequence of continuous random variables such that:
then prove that .