Cochin University of Science and Technology

Prove sinc(t) is in L1.

Statistical signal processing

Consider a sequence of independent and identically distributed trials, say X1,X2,··· ,Xn... of a Bernoulli random variable X ∼ Bern(p), where ∈ [0,1]. Describe in words the event which characterizes the following random variables (look up their definitions): (a) Binomial random variable (b) Geometric random variable (c) Negative binomial random variable (d) Hypergeometric random variable (e) Pascal random variable (f) What is the probability for the following events arising out of {Xi}: • the first success does not occur till the m-th trial (m is a constant) • the third success occurs between the m-th and m+k-th trial (m and k are constants) (g) Let m1,m2··· ,mk be fixed and given numbers. Let Y denote the number of trials for the k+1-th success after the 1-st success occurs at m1, the second success occurs at m2, and so on till the k-th success occurs at mk. Find the range of values taken by Y and the distribution of Y. (h) Let Z denote the number of trials needed for the first occurrence of 3 successive ones (1s), i.e., to get the f irst run-of-1s of size 3. What is the distribution of Z? (i) Create an interesting problem of your own with Bernoulli trials?

An item is produced in three factories AA, BB and CC. Factory AA produces 2 times the number of items produced by factory BB, and the factories BB and CC produces the same number of items. It is known that 8%, 7%, 3% of the items produced by factories AA, BB and CC respectively are defective. All items produced in the three factories are stocked, and an item is selected at random. If an item selected at random is found to be defective, what is the probability that it was produced by factory B? (Enter the answer correct to two decimal places)

An item is produced in three factories AA, BB and CC. Factory AA produces 2 times the number of items produced by factory BB, and the factories BB and CC produces the same number of items. It is known that 8%, 7%, 3% of the items produced by factories AA, BB and CC respectively are defective. All items produced in the three factories are stocked, and an item is selected at random. What is the probability that the selected item is defective? (Enter the answer correct to two decimal places)

A and B predicts the outcomes of a cricket match and their chances of predicting the runs scored by a specific batsman correctly are 1/6​ and 3/6​ respectively independent of each other. If the probability of them predicting the same wrong score is 2/207​. Given that they predicted the same score, find the probability that their answer is correct. a)205/207 (b) 30/207 (c) 621/651 (d) 30/651