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 those that are discussed in SLM. If you denote your created random variable by X then find the probability that X is less than 2.

 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.

If X ~ Gamma , and Y ~ Gamma , ( ) be two independent gamma distributions and  and V = X+ Y  then find the distribution of U.

A hospital specialising in heart surgery. In 2022 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 25000. 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) Maximum amount paid by the lowest paying one-third of patients.

A hospital specialising in heart surgery. In 2022 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 25000. 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) Maximum amount paid by the lowest paying one-third of patients

If X ~ Gamma , and Y ~ Gamma , (θ α) (θ β) be two independent gamma distributions and

    and    then find the distribution of U. 

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 X₁,X₂,X_3,X₄,........be a sequence of continuous random variables such that f_x2(x)=then prove that