edx S G t l pt | | | UBC | UBC OF G G se BjMI G UBC G X C File | C:/Users/bryan/Downloads/STAT251_Assignment1_21W1.pdf Apps Google MET f (18) Facebook YouTube cox ER12[ |edX P MATH 104 ALL 20... Dashboard UBC webmail UBC UBC Student Servi... > > Reading list E STAT251_Assignment1_21W1.pdf 2 / 4 - 76% + Question 3 (a) Suppose you are outside a polling station during a local election and asking people if they voted for a specific independent candidate, John Smith. Consider the answer you get from each person follows a Bernoulli distribution with success rate 7 E (0, 1), where success represents the event that a person voted for John Smith. Then, the geometric distribution would represent the variable X, which is the number of people you had to poll before you find someone who voted John Smith: p(X = k) = P(polled k persons before find someone voted for John) = (1-7)*n, VK E N. 2 i) Without computing any individual probability associated to each value that the random variable X could take on, show that the function above fulfills the nec- essary conditions in order to be defined as a probability mass function pmf (i.e. p(k) 2 0 and Ek=op(k) = 1). (4 marks) ii) Suppose 7 = 0.3. Using the cdf, what is the probability that at least the first 3 persons did not vote John Smith? Write the proper notation for defining the probability. Keep 3 decimal places. (2 marks) 3 LINE 12.C F+ 06:54 150 2021/10/20edx S G t l pt | | | UBC | UBC OF G G se BjMI G UBC G X C File | C:/Users/bryan/Downloads/STAT251_Assignment1_21W1.pdf Apps Google MET f (18) Facebook YouTube ex ER12 0 | edX P MATH 104 ALL 20.. Dashboard UBC webmail UBC UBC Student Servi... > > Reading list E STAT251_Assignment1_21W1.pdf 3 / 4 | - 76% + i) Without computing any individual probability associated to each value that the random variable X could take on, show that the function above fulfills the nec- essary conditions in order to be defined as a probability mass function pmf (i.e. p(k) 2 0 and Ex-OP(k) = 1). (4 marks) ii) Suppose 7 = 0.3. Using the cdf, what is the probability that at least the first 3 persons did not vote John Smith? Write the proper notation for defining the 2 probability. Keep 3 decimal places. (2 marks) (b) Suppose the number of minutes someone spends inside the Birdcoop gym is distributed Exponentially with mean 30 minutes - that is, follows a Exponential( 30) distribution. Assume that each person's time spent at the gym is independent from each other. Note that if X ~ Exponential()) then the CDF is given by P(X = 30 as in part a). Round your answer to 3 decimal places. (2 marks) Question 4 You are looking for a student at your college who lives within 10 kilometers from you. You recall that 70% of the students live within 10 kilometers from you. You randomly contact students from the university until one says he or she lives within 10 kilometers from you. Assume that there are an infinite number of students you can contact at your college. (Round 12.C F+ 06:54 50 LINE 2021/10/20edx S G t l pt | | | UBC | UBC OF G G se BjMI G UBC G X C File | C:/Users/bryan/Downloads/STAT251_Assignment1_21W1.pdf Apps Google MET f (18) Facebook YouTube cox BRAZEN |edX P MATH 104 ALL 20... Dashboard UBC webmail UBC UBC Student Servi... > > Reading list E STAT251_Assignment1_21W1.pdf 3 / 4 | - 76% + Question 4 You are looking for a student at your college who lives within 10 kilometers from you. You recall that 70% of the students live within 10 kilometers from you. You randomly contact students from the university until one says he or she lives within 10 kilometers from you. Assume that there are an infinite number of students you can contact at your college. (Round your answer to 4 decimal places) 2 (a) What is the probability that you needed to contact exactly four people to find the first one who lives within 10 kilometers from you? Please define the random variable and write down the probability function. (3 points) (b) What is the probability that you have to contact at least 3 people to find the first one who lives within 10 kilometers from you? (3 points) 3 (c) Suppose every person you call will pick up their phone and answer your question. Also, suppose that each call lasts 30 seconds. Calculate the expected time you will spend on the phone as well as its standard deviation. (4 points) LINE w 12.C F+ 06:54 50 2021/10/20edx S G t l pt | | | UBC | UBC OF G G se BjMI G UBC G X C File | C:/Users/bryan/Downloads/STAT251_Assignment1_21W1.pdf Apps Google Ali f (18) Facebook YouTube edx 12121 | edX P MATH 104 ALL 20.. Dashboard UBC webmail UBC UBC Student Servi... > > Reading list E STAT251_Assignment1_21W1.pdf 2 / 4 - 80% + midterm and final using the plot. (4 marks) Question 2 (a) (6 marks) After having some lunch in KFC, you drive to IKEA for buying some bowls, however you lost your keys either in the KFC or in IKEA. The probability for loosing it at either location is 1/2 . The probability that you will find your keys if you lost it in IKEA is 30%, while the probability that you will find it if you lost it at the KFC is 70%. i) Please specify the probability that you will not find your keys in the KFC? Define all the events carefully needed in the calculation. (3 marks) ii) You went to KFC but you could not find your keys there. Given this information, what is the probability that you will find it in IKEA? (3 marks) (b) (6 marks) Last century, the annual income of farmers in a country are normally dis- tributed with mean of 500$ and standard deviation of 200$. i) Calculate the probability that a randomly selected farmer's income is between 500$ and 520$.(2 marks) 2 ii) If the probability that the annual income of farmers in this county is within (500-a , 500+a) is not less than 0.95, how much at least a is? (2 marks) iii) Calculate the probability that the sum of 25 randomly selected farmers' income will be more than 13,500$. (2 marks) Question 3 (a) Suppose you are outside a polling station during a local election and asking people if 150 LINE W 12.C F + 06:54 2021/10/20