Let (Ω, , ) be a probability space and  be a sequence of independent and identically distributed (i.i.d.) random variables from the uniform distribution on the interval [12, 20]. If Xn denotes the sample mean of the first n random variables of the sequence

 and  Find the value of a.

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.

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?

 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 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 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.

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

    and    then find the distribution of U.