Step by step workings would be very much helpful! Thank you:)

1. A 5-digit number is formed such that each digit is one of the nine integers 1; 2; :::; 9. Each integer can be used any number of times. (i) How many 5-digit numbers can be formed? (ii) How many 5-digit numbers can be formed such that no three consecutive digits are the same? 2. The system contains four gates and water ows from A to B. The gates work independently and the probability that a gate is open is p. B (i) What is the probability that water is able to ow from A to E? (ii) What is the conditional probability that gate 2 is open given that water is able to flow from A to B? 3. Two points are selected randomly on a line of length 1. as X and Y as shown in the gure. Therefore X is the smaller of the two points and Y is the larger of the two points. 0 X Y 1 (i) Find the joint probability density function of X and Y . (ii) What is the expected length E{Y - X)? 4. Jar A contains 2 tags numbered 1 and 2 respectively- Jar B also contains 2 tags numbered 1 and 2 respectively. Jar G contains 3 tags numbered 1, 2 and 3 respectively. We select 1 tag at random from A. Record the number as X1. We then select randomly X1 tags from B with replacement. Let X2 be the total number on the tags drawn from B. We nally select X2 tags randomly from C with replacement. Let X3 be the total number on the tags drawn from 0. Find E{X3}. 5. Let X and Y be independent standard normal random variables. Calculate the conditional expectation EtXa-Y an-Y =1). Hint: X + Y and X - Y are independent. You may assume this fact without proof