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 _____ ___ __________ ______ ______ _____ _________ ___.
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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.
Explain the procedure of assigning probability in a continuous world of probability theory.
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 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.
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.
If is a sequence of continuous random variables such that:
then prove that .