Topic 3 – Discrete p.d.f - Math'scool

1 Topic 3 Discrete 1. Spec 1992 / S1(new) - Qu 6: (a): Find E(3X-2Y). A Discrete random variable X has : (b): Form (as a table) the of Z where Z = XY. (c): State E(Z). p( x) = kx 2 x = 1, 2, 3. 5. Jun 94 / S1 - Qu 2: p ( x) = 0 otherwise The random variable X has : Find: (a): k x 0 1 2 3. (b): E(X) P(X=x) (c): Var(X). (d): Find E(2X-4) and Var(2X-4) (a): Find . Form the of X and hence find: 2. Jan 92 / S1(new) - Qu 6: (b): p(1 < X 3). The random variable X has : (c): F( ). p( X = x) = cx x = 1, 2.

2 , 6 6. Jan 95 / S1 - Qu 5: The Discrete random variable X has : p( X = x) = 0 otherwise p( X = x) = kx x = 1, 2, , .., 5. Find the values of: (a): c p( X = x) = 0 otherwise (b): E(X). (c): Var(X) (a): Show that k = 1/15. Y is an independent Discrete where E(Y) = and Var(Y) = 2 (b): Find E(X+3). (d): Find E(2X-Y) and Var(2X-Y) (c): Find Var(2X-4). (e): Find E(2X +Y) 7. Jun 95 / S1 - Qu 3: For the random variable X, E(X) = 5 and E(X ) = 36. Find: 3. Jun 93 / S1 - Qu 4: (a): E(3X + 2X + 1). X has : (b): Var(X).

3 (c): Var(3X). x -1 0 1 2. p(X x) 1 8. Jan 96 / S1(old) - Qu 4: The Discrete random variable X has : (a): Find p(-1 X < 1). (b): Find E(2X+3) kx Two independent observations X1 & X2 are taken from X p ( x) = x = 2, 3. The random variable Y represents the sum of these observations x +1. 2. (c): Find p(Y = 0) 2kx (d): Form (as a table), the of Y p ( x) = 2 x = 4, 5. x 1. 4. Jan 94 / S1 - Qu 2: p ( x) = 0 otherwise If X is a Discrete random variable with : (a): Show k = 20/33. p ( X = x) = 6x x = 1, 2, 3 (b): Find the probability that X takes a value less than 3 or greater than 4.

4 P( X = x) = 0 otherwise and Y is a Discrete random variable with : 9. Spec 95 / T1(new) - Qu 1: 2. The random variable X has : p (Y = y ) = y5 x = 1, 2 x 1 2 3 4. p(Y = y ) = 0 otherwise P(X=x) 1/4 1/2 p 1/6. (a): Find p (a): Form the of X. (b): Find = E(X) (b): Find E(X). (c): Find p(X < ) (c): Show Var(X) = 179/12. If the indicator moves upwards over time then the economy is growing 10. Jun 95 / T1(new) - Qu 8: Otherwise the economy is declining A & B are independent Discrete random variables and have : (d): Comment on the state of the economy predicted by the model a 1 2 3 b 1 2 Before the stock market opened on Monday morning, the indicator had P(A=a) 1/4 1/2 1/4 p(B=b) 1/3 2/3 a value of 3373.

5 (e): Use the model to predict the probability that the indicator is The random variable Q is the product of one observation from A and one at least 3400 when the market closes on Friday that week from B. The student feels that this is too low and suggests the increase in (a): Show p(Q = 2) = 1/3. the indicator over 5 days could have a normal distribution with a mean (b): Form the for Q. of and a variance of 895/12. (c): Hence show E(Q) = 10/3. (f): Find the prob the indicator is at least 3400 using this model (d): Find Var(Q).

6 14. Jun 98 / T1 - Qu 5: 11. Jan 96 / T1 - Qu 9: The Discrete random variable X has : Plumbing students know that pipes are only made in 2 cm, 5 cm 8 cm, 15 cm and 20 cm diameters x 1 2 3 4 5. The teacher asked 50 students to guess the diameter of a pipe that he P(X=x) knew was 8 cm and these were the results: Find (a): p(2 < X 4) (b): F( ). Diameter 2 cm 5 cm 8 cm 15 cm 20 cm (c): E(X) (d): Var(X). Frequency 2 9 27 11 1 (e): E(X +4X-3). The trainer wants to model the error made by a random apprentice 15.

7 Jan 99 / T1 - Qu 6: So the X is the apprentices estimate minus 8 When a cell is subjected to radiation the cell may die, survive as a (a): Form a for X single cell or divide into two cells with probabilities 1/2, 1/3 & 1/6. (b): Show E(X) = 1 Two cells are independently subjected to radiation (c): Find Var(X) The random variable X represents the total number of cells in The teacher also wants to model the errors the apprentices make in existence after the experiment estimating the length of an 8 cm wire (a): Show p(X = 2) = 5/18.

8 (d): Explain why the X is not a suitable model in this case (b): Form the of X. (c): Find E(X). 12. Jun 96 / T1 - Qu 4: (d): Show Var(X) = 10/9. The Discrete random variable X had : Another two cells are submitted to radiation in a similar experiment x 0 1 2 3 4 and the random variable Y represents the number of cells in existence p(X x) after this experiment The random variable Z is defined as Z = X - Y. (a): Write down the name of this distribution (e): Find E(Z) and Var(Z). (b): Find p(0 X < 2).

9 (c): Find E(X) 16. Jun 99 / T1 - Qu 2: (d): Find E(X +3x) The random variable X has a mean of 5 and a variance of 5. The random variable Y has : 13. Jan 98 / T1 - Qu 8: A student is trying to model the daily movement X points of the stock market p (Y = y ) = 1 5 1, 2, 3, 4, 5. He assumes the movement of X on any day is independent of the next A die is rolled and if an odd number is shown then the indicator is p(Y = y ) = 0 otherwise moved up that number of points Find (a): E(3X-2Y). If an even number is shown then the indicator is moved down that (b): Var(3X-2Y).

10 Number of points (c): What assumption have you made 17. Jan 00 / T1(old) - Qu 3: 21. Mock 00 / S1(new) - Qu 4: The Discrete random variable X has : A customer has to type in a PIN number to withdraw money If the wrong number is typed in, the customer can try again up to a x 1 2 3 4 5 maximum of 4 attempts in total P(X=x) a a b The probability of success at each attempt is Given E(X)=3, find: (a): Show the prob of getting correct number at 3rd attempt is (a): a & b The A represents the number of attempts made, regardless of (b): Var(X) whether or not the attempt is successful (c).