9. Let X be a random variable with expected value 2 and variance 3. Compute the following quantities: a) E(2X+3) b) Var (2X) c) Var (X+3)

В2. (a) Given the probability Venn diagram below for the two Events A and B find the probabilities: 0.6 0.5 A 0.2 (i) (ii) (iii) Pr( A 'OR' B) Pr( Not-A 'AND' Not-B) Pr( Not-A "OR' Not-B) (b) A random sample from a population of X with unknown mean but a known variance, o = 400, gives the following data: sample size, n= 125, Ex= 1750. (i) Give a point estimate of the population mean, u. (ii) Is this estimate biased or not. Give a reason. (ii) Derive a 95% confidence interval for the population mean, µ.

2 (a) Let Y be the random variable that counts the number of sixes which occur when a die is tossed 10 times. What type of random variable is Y? What is P (Y = 3), expected number of sixes,Var(Y )? (b) Let U be the random variable that counts the number of accidents that occur at an intersection in one week. What type of random variable is U? Suppose that, on average, 2 accidents occur per week. Find P(U = 2), E(U), and V ar(U). (c) Let X be the recorded body temperature of a healthy adult in degrees Fahrenheit. What type of rv is X? Estimate its mean and standard deviation, based on your knowledge of body temperatures.

We have a random variable X1 for population 1 with a mean value of 52 and standard deviation of 3. we have a random variable X2 for population 2 with a mean value of 50 and population standard deviation of 4. we obtain a random sample of 625 elements independently for both populations. what is the probability that the difference in the sample averages X1-X2 is at least 2.3?