The following circuit operates if and only if there is a path of functional devices from left ta right. The probability of each device functions is as shown. Assume that the probability that a device functions does not depend on whether or not other devices are functional. What is the probability that the circuit operates Round your answer to four decimal places (e.g. 98.7654).

12) The board of examiners that administers the real estate broker's examination in a certain state found that the mean score on the test was 513 and the standard deviation was 72. If the board wants to set the passing score so that only the best 80% of all applicants pass, what is the passing score? Assume that the scores are normally distributed.

1.Let X be a continuous random variable with the probability density function f(x)=k \cdot \sqrt{x}, \quad 0 \leq x \leq 4 \begin{aligned} & f(x)=0, \\ & \text{^^20otherwise.^^20}\end{aligned} a)What must the value of k be so that f(x) is a probability density function? b)Find the cumulative distribution function of X, Fx(x)=P(X<=x). c)Find the probability P(1<= X <=2). d)Find the median of the distribution of X. That is, find m such that P(X<=m)=P(X >=m) =½. e) Find the 30th percentile of the distribution of X. \text { g) Find } \sigma_{X}=\operatorname{SD}(X) \text { . } \text { f) Find } \mu_{X}=E(X) \text { . }

(a) Show that Mn = ((1 – p)/p)Sn is a Martingale. (c) Show that M, = Sn – (1 – 2p)n is a Martingale. (d) Let Tn = min{n,T} and show that M = MT, is a Martingale. (e) Use optional sampling to estimate E(T).

Prior to 2010, standard licence plates in Alberta consisted of 3 letters followed by 3 digits, and standardOntario licence plates consisted of 4 letters followed by 3 digits. Letters and digits can be repeated onlicence plates. There are 23 letters in the alphabet that can be used, as the letters I, O, and Q are notallowed. All of the digits from 0 to 9 may be used. An example of each licence plate is shown below. In 2009, the number of standard licence plates possible in Ontario was r times greater than the number of standard licence plates possible in Alberta. The value of r as a whole number is[1]

15) Farmers often sell fruits and vegetables at roadside stands during the summer. One such roadside stand has a daily demand for tomatoes that is approximately normally distributed with a mean of 131 tomatoes and a standard deviation of 30 tomatoes.How many tomatoes must be available on any given day so that there is only a 1.5%chance that all tomatoes will be sold?

2. The total number of cards in a deck of Italian playing cards is 40. Out of those 40 playing cards there are 4cards that are a King. Consider an experiment where the deck is properly shuffled and a single card is dealt. (a) What is the probability that the card dealt is aKing? (b) Now suppose that the card is returned to the deck and that the deck is once again properly shuffled. The experiment is repeated another 19 times (i.e. in total 20 trials of the experiment are conducted - including the initial trial in (a) above) where the card that is dealt is returned to the deck after each trial and the deck is properly shuffled again before the next card is dealt. Out of the 20 trials, let X equal the number of King cards that are observed. i. What are all of the possible values that can be observed for X? ii. What is the expected frequency for the number of King cards dealt? Show workings. iii. Suppose that X = 4 is observed. What is the relative frequency for this many King cards dealt?(2 тarks) iv. Now suppose that the experiment is to be repeated but this time with a very large n (number of trials). For a very large n, what should the relative frequency for the number of King cards dealt be approximately equal to?(1 mark)

Question 2 In a particularly addictive game (Helicopter Game) you have to get a helicopter to fly some distance in arbitrary un Suppose the distance you travel can be modeled as a random variable D having pdf f(d) = 1.2 (1+0.4d) d>0 else Where d is measured in thousands (1= 1000 units) 1. Determine the 90th percentile distance traveled (i.e. find k such that P(D≤k) = 0.9). 2. Determine the expected distance traveled on a single run. 3. Determine the variance and standard deviation of the distance traveled. 4. If you have traveled 1500 units so far what is the probability you will travel more than 2000 units total? 5. How many rounds can you expect to play in order to have a round where you travel more than 5000 units? 6. What assumptions must you make in (5)?

(a) The probability is _____________. (b) The probability is___________. (c) The probability is___________. The amount of time that a customer spends waiting at an airport check-in counter is a random variable with mean 8.2 minutes and standard deviation 1.3 minutes. Suppose that a random sample of = 50 customers is observed. Find the probability that the average time waiting in line for these customers is (c) Less than 6 minutes Round your answers to one decimal place (e.g. 98.7). (b) Between 5 and 10 minutes (a) Less than 10 minutes

6. What is the probability that X is between 15 and 25 (inclusive)? What is the exact answer(use the pmf table)? What would the normal approximation predict (You should do this calculation by hand)? How close are these?