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Probability and Statistics EQT 272 Semester 2 2013/2014 TUTORIAL 2 1) Determine whether the following random variables are discrete or continuous. i. The number of eggs that a hen lays in a day. ii. The amount of milk a cow produces in one day. iii. The cost of making a randomly selected movie. iv. The number of goals scored by a randomly selected football player in a soccer tournament. 2) Determine the value c so that the following function is a probability function for a discrete random variable. f ( x) c( x2 5 ) for x 0,1,2,3,4 2 2 ans: 2/55 3) A box contains three marbles (one blue, one red and one yellow). Two marbles are drawn with replacement. This means a marbles is selected, its colour is observed and then it is replaced in the box. A second marble is then selected and its colour is observed. Let B denotes “blue” , R denotes “red” and Y denotes “yellow”. i. List the possible outcomes (the elements in the sample space S) ans: S = {BB, BR, BY, RB, RR, RY, YB, YR, YY} ii. Let X be a random variable giving the number of “yellow” marbles. List the outcomes for the random variable X. ans: X={0, 1, 2} iii. Find the probability for each value of X. ans: 4/9, 4/9, 1/9 4) A factory manufactures DVDs. Batches of DVDs are randomly selected. The number of defects (X) for each batch is observed and the following distribution is obtained. X 0 1 2 3 4 5 P(X=x) 0.502 0.365 0.098 0.023 0.011 0.001 i. Verify whether this distribution is a probability distribution. ii. Find P(X ≥ 2) ans: 0.133 iii. Find P(0<X<4) ans: 0.486 Probability and Statistics EQT 272 Semester 2 2013/2014 iv. Find the E(X), Variance(X) and standard deviation X. 5) Let the following function of a random variable Y be 0 y 1 1 y 2 otherwise y, f ( y ) 2 y , 0 i. Check whether the function is probability density function ii. Find the cumulative distribution function of Y, F (Y) iii. Find P(0.5≤Y≤0.9) ans: 0.28 iv. Find P(0.75≤Y≤1.5) ans: 0.594 6) If a random variable X has probability density function B x 2 4 x 5 f ( x) 0 calculate the value of B. 0 x5 x 0 or x 5 ans: 3/50 7) Suppose X is a random variable with the following distribution function. x0 0, 3 x 3 F(X ) (4 x 2 ), 0 x8 3 256 x 8 1, Find P(-1≤ X ≤ 2) i. ans: 0.156 ii. Find f (x) 8) Let Y be a continuous random variable with the following probability density function. 0 y 1 1 y 2 otherwise y, f ( y ) 2 y, 0, Calculate i. expected value of Y, E(Y) ans: 1 ii. variance of Y, Var (Y) ans: 1/6 iii. Y ans: 0.408

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