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Question 1 Question 2 Question 3 Question 4 Question 5 Math 144 tutorial 4 March 8, 2010 Question 6 Question 7 Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 1. The probability density function of a random variable X is given by: √ k , if − 1 < x < 0; 1 − x2 (1) f (x) = 0, elsewhere. (a). Calculate the value of k, (b). Find the probability distribution function of X. Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 1. The probability density function of a random variable X is given by: √ k , if − 1 < x < 0; 1 − x2 (1) f (x) = 0, elsewhere. (a). Calculate the value of k, (b). Find the probability distribution function of X. Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 2. Let X denote the lifetime of a radio, in years, manufactured by a certain company. The density function of X is give by: 1 e− 15x , if x ≥ 0; (2) f (x) = 15 0, elsewhere. What is the probability that, of eight such radios, at least four last more than 15 years? Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 2. Let X denote the lifetime of a radio, in years, manufactured by a certain company. The density function of X is give by: 1 e− 15x , if x ≥ 0; (2) f (x) = 15 0, elsewhere. What is the probability that, of eight such radios, at least four last more than 15 years? Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 3. First a point Y is selected a random from the interval (0,1). then another point X is selected at random from the interval (Y, 1). Find the probability density function of X. Question 7 Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 3. First a point Y is selected a random from the interval (0,1). then another point X is selected at random from the interval (Y, 1). Find the probability density function of X. Question 7 Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 4. Let (X, Y) be a random point from a unit disk centered at the 4 origin. Find P(0 ≤ X ≤ 11 |Y = 45 ). Question 7 Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 4. Let (X, Y) be a random point from a unit disk centered at the 4 origin. Find P(0 ≤ X ≤ 11 |Y = 45 ). Question 7 Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 5. Stores A and B, which belong to the same owner, are located in two different towns. If the probability density function of the weekly profit of each store, in thousands of dollars, is given by ( x/4, if 1 < x < 3; f (x) = (3) 0, otherwise. and the profit of one store is independent of the other, what is the probability that next week one store makes at least $500 more than the other store? Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 5. Stores A and B, which belong to the same owner, are located in two different towns. If the probability density function of the weekly profit of each store, in thousands of dollars, is given by ( x/4, if 1 < x < 3; f (x) = (3) 0, otherwise. and the profit of one store is independent of the other, what is the probability that next week one store makes at least $500 more than the other store? Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 6. A random variable Y with distribution function F(y) = y/5 for 0 < y ≤ 5. Determine the conditional expectation of Y, E[Y|Y > x] give that Y > x and 0 < x ≤ 5. Question 7 Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 6. A random variable Y with distribution function F(y) = y/5 for 0 < y ≤ 5. Determine the conditional expectation of Y, E[Y|Y > x] give that Y > x and 0 < x ≤ 5. Question 7 Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 7. In a given lottery, players pick six different integers between 1 and 49, the order of selection being irrelevant. The lottery commission then select six of these numbers at random as winning numbers. A player wins the grand prize of $1,200,000 if all six numbers that he has selected match the winning numbers. He wins the second and third prizes of $800 and $35, respectively, if exactly five and four of his selected numbers match the winning numbers. What is the expected value of the amount a player wins in one game? Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 7. In a given lottery, players pick six different integers between 1 and 49, the order of selection being irrelevant. The lottery commission then select six of these numbers at random as winning numbers. A player wins the grand prize of $1,200,000 if all six numbers that he has selected match the winning numbers. He wins the second and third prizes of $800 and $35, respectively, if exactly five and four of his selected numbers match the winning numbers. What is the expected value of the amount a player wins in one game?
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