A LEVEL MATHEMATICS QUESTIONSHEETS DISCRETE RANDOM VARIABLES

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1 1. Two fair dice are thrown. The random variable T represents the maximum score on the two dice. (For example, if the first dice shows a and the second a 4, then the value of T would be 4). a) Find the probability distribution of T b) Find E(T) c) Find E(T 2 ) d) Hence find Var(T) 2. The random variable X has the following probability distribution: Find x P(X=x) 1 α 2 2α a) α b) The expectation of X c) The standard deviation of X. A bag contains red balls and 2 green balls. Three balls are taken from the bag without replacement. The discrete random variable R is the number of red balls drawn. a) Show that P(R = 1) = 10 b) Find the probability distribution of R c) Find E(R) [] pg. 1

2 4. The discrete random variable X takes values from 2,, 4 and 6 with probabilities given by: P(X = x) = x k x = 2,, 4, 6 a) Show that k = 5 4 b) Calculate E(X) c) Calculate Var(X) d) Calculate F(.1) 5. A game at a fair involves shooting at a target, as shown below. Each player had one shot at the target. bull outer ring target The radius of the bull at the centre of the target is 2cm. The radius of the outer ring is 12cm. The radius of the whole target is 0cm. In order to work out the amount that should be given in prizes, the owner of the game decides to model the situation as follows : All shots hit the target The probability a shot hits a particular region of the target is proportional to the area of that region. Using this model : 7 a) Show that the probability of hitting the outer ring is 45 b) Find the probabilities for hitting the bull and the unshaded area of the target. [] The owner decides to award a prize of for hitting the bull, and 1 for hitting the outer ring. He charges each customer 25p. c) Find the expected profit the owner makes per player, giving your answer in pence correct to 2 decimal places. d) Comment on the validity of the model. pg. 2

3 6. A 5-sided spinner has the numbers 1-5 on its sides. The random variable X is the value shown on the spinner. Assuming the spinner is fair: a) Suggest a suitable model for X b) State the name of the distribution you have suggested in a) c) Use this model to find the mean and standard deviation of X. [] When the spinner is spun 1000 times, the mean value of the score is.8 and the standard deviation is 1.0. d) Comment on the validity of the model. [] e) Explain how an improved model could be devised. 7. A salesman has a probability of 0.1 of making a sale at each house he knocks at. The discrete random variable X is the number of houses he has to call at unsuccessfully before making his first sale. (So if he made a sale at his first house, X would be zero.) f(x) is the probability function for X. a) State the value of f(0) b) Calculate f(2) c) Calculate F(2.1) [] pg.

4 8. The discrete random variable Y has probability function f(y), as defined below. 1 f(y) = 8 y is an integer, 5 y k a) State the value of k. b) State the name of this distribution. The discrete random variable X is defined as follows : If Y 9, X = If Y 6, X = 6 Otherwise, X = 2. c) State P(X = ) d) Calculate the expectation of X 9. The random variables A and B are independent, and are distributed as shown below. a 1 5 b 1 5 P(A = a) P(B = b) a) State E(A). b) Calculate E(B). The random variable X is the product of random variables A and B. c) Show that P(X = ) = 6 1 d) Write down the probability distribution of X e) Find E(X), and verify that E(X) = E(A)E(B) [7] [] pg. 4

5 10.A discrete random variable T takes values from 0 to 5 with probabilities given by: 2t + 1 t = 0,1, 2, 20 P(T = t) = 11 2t t = 4, 5 20 a) Find E(T) b) Find Var(T) c) Write down the values of i) E(2T +1) ii) Var(2T + 1) [] 11. The random variable X takes values 1, 2,, 4 with probabilities defined below : x P(X = x) α α 2α 2α a) Find the value of α. b) Calculate E(X) and E(X 2 ). c) Hence or otherwise, find : i) E(2X ) ii) E(X 2 X 1) iii) Var(2X) pg. 5

6 12. The independent discrete random variables X and Y have probability distributions as shown below: x y P(X = x) P(Y = y) a) Calculate E(X) and E(Y). b) State the value of E(2X Y). [] The random variable Z is defined as follows : Z = 1 if Y X = 2 Z = 0 otherwise c) Find P(Z = 1), and hence find E(Z). [6] 1. Angela can travel by bus or train to work in the morning. The bus fare is 60p, and the train fare is 90p. If she catches the bus, she has a probability of 0.4 of being late, but if she catches the train she has a probability of 0.2 of being late. If Angela is late to work, she is fined 1 from her wages. a) Show that Angela s expected morning total cost if she catches the bus is 1, and find her expected morning cost if she catches the train. b) Each working morning, Angela catches the train with probability p. Given that her expected morning cost is 1.08, find the value of p. pg. 6