**Subject Code & Title :** ST104a Statistics

In this exercise you will practise probability problems, random variables and the normal distribution. Questions 1 and 2 require the total probability formula and Bayes’theorem. Question 3 requires the derivation of a discrete probability distribution. Question 4 involves the calculation of the expected value and variance for a discrete random variable. Questions 5 to 7 work with the normal distribution – be sure to distinguish between working with X and X ̄.**ST104a Statistics Assignment Exercise 2 – UK.**

1.Suppose there are two boxes. The first box contains three green balls and one red ball, whereas the second box contains two green balls and two red balls. First, a box is chosen at random and then a ball is drawn randomly from that box.

(a) What is the probability that the ball drawn is green?

(b) If the ball drawn was green, what is the probability that the first box was chosen?

2.In an introductory economics class, the numbers of males and females are 16 and 24, respectively.

(a) A student is selected randomly from the class. What is the probability the student is female?

(b) A student is selected at random and removed from the class. A second student is then selected. What is the probability that one of the two students is male and the other is female?

(c) What is the probability that the second student is male, given that the first student is female and removed from the class?

(d) In previous years it was found that 80% of males pass the examination and 85% of females pass. Based on the available information, find the probability that a student who passes the examination is female.

**ST104a Statistics Assignment Exercise 2 – UK.**

3.Two fair dice are thrown.

(a) Suppose that M denotes the largest of the scores on the two dice. Determine the probability distribution of M.

(b) You are told that the sum of the scores on the two dice is at most 4. What is the probability of at least one score being 2?

4.The probability distribution of a random variable X is given below.

(a) Find the probability that X is larger than 2.

(b) Find the expected value of X, E(X).

(c) Find the variance of X, Var(X).

5.A test is taken by some students. Assume that the marks follow a normal distribution with a mean of 60 and a variance of 25.

(a) Calculate the probability that the mark of a randomly selected student is greater than 58.

A random sample of size n = 5 is selected.

(b) Calculate the probability that the sample mean lies between 59 and 61.

6.House hold expenditure in country A is normally distributed with a mean of $1,200 per week and a standard deviation of $400 per week. In country B it is also normally distributed but with a mean of $960 per week and a standard deviation of $200 per week. Which country has a higher proportion of households spending less than $800 per week?

**ST104a Statistics Assignment Exercise 2 – UK.**

7.Suppose that X is a normally distributed random variable with a mean of 0 and a variance of 1.

(a) Find the probability that X + 4 is less than 4.

(b) Find the value of b so that the probability of X −b being less than zero is 0.975.

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