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- Suppose a particle moves along tje x-axis beginning at 0. It moves one integer step to the left or right with equal probability. What is the probability function of its position after four steps?Daniel is 1000 meters from his home and decides to walk home. Every time he takes one step, he goes forward 1 meter with a probability of 2/3 and backwards 1 meter with a probability of 1/3, regardless of the steps taken earlier. (a) Calculate the expected value of the distance to his home after 1000 steps. (b) Calculate approximately the probability that Daniel, after 1000 steps have come at least 300 meters closer to his home. Plz explain.Two drunk start out together at the origin, each having the equal probability of making a step to the left or right along the x-axis. Assume that men make their steps simultaneously. Let N be the total number of steps taken by each man. Assume that a particular instant, they are separated by some distance. In this situation, in a given particular step. If they move towards each other, the separation between them decreases. If the move away from each other, the separation between them increases If they move in the same direction, the separation remains unaltered Then calculate (a) The probability of decreasing their separation (b) The probability of increasing their separation (c) The probability of no separation between them Out of N steps, n1 steps belong to decreasing the separation of them; n2 steps belong to increasing the separation of them and n3 steps belong to no separation. Then, calculate the following. (d) The probability of taking the steps n1, n2 and n3 (e) The…
- Two drunk start out together at the origin, each having the equal probability of making a step to the left or right along the x-axis. Assume that men make their steps simultaneously. Let N be the total number of steps taken by each man. Assume that a particular instant, they are separated by some distance. In this situation, in a given particular step. If they move towards each other, the separation between them decreases. If the move away from each other, the separation between them increases If they move in the same direction, the separation remains unaltered Out of N steps, n1 steps belong to decreasing the separation of them; n2 steps belong to increasing the separation of them and n3 steps belong to no separation. Then, calculate the following. (a) The probability of taking the steps n1, n2 and n3 (b) The probability that they meet again at N stepsA frog starts at location A and wants to get to location B that is 500 meters away.Starting at A, it sits at a given spot on the path from A to B for a time that is exponen-tially distributed with parameter 1 (minute). After that, it jumps 1 meter towards B, sitsin the new spot for an exponentially distributed time, etc. All the times are independent.What is the probability (approximately) that it will get to B within 7 hours?The random variable x follows a uniform probability distribution in the interval (0,1); the probability of obtaining x values outside this range is zero. What is the probability distribution of y = ln x?
- According to the Maxwell–Boltzmann law of theoret-ical physics, the probability density of V, the velocity of a gas molecule, isf(v) =⎧⎨⎩kv2e−βv2for v > 00 elsewhere where β depends on its mass and the absolute tem-perature and k is an appropriate constant. Show that the kinetic energy E = 1 2mV2, where m the massof the molecule is a random variable having a gammadistribution.Andrew, who is planning to retire at age 60, is analyzing various pension options for himself and his spouse, Kim. According to actuarial tables, the probability that Andrew will live for at least 20 years after retiring at age 60 is 63%. Kim will be 57 years old when Andrew retires. The probability that Kim will live for at least 20 years after that is 77%. Assume that the life span of Andrew has no effect on the life span of Kim. (In other words, assume that whether Andrew is alive in 20 years and whether Kim is alive in 20 years are independent events.)a) What is the probability that Andrew does not live for at least 20 years after retiring?b) What is the probability that Kim does not live for at least 20 years after Andrew retires?c) What is the probability that both Andrew and Kim live for at least 20 years after Andrew retires?d) What is the probability that neither Andrew nor Kim lives for at least 20 years after Andrew retires?e) What is the probability that either Andrew or…Andrew, who is planning to retire at age 60, is analyzing various pension options for himself and his spouse, Kim. According to actuarial tables, the probability that Andrew will live for at least 20 years after retiring at age 60 is 63%. Kim will be 57 years old when Andrew retires. The probability that Kim will live for at least 20 years after that is 77%. Assume that the life span of Andrew has no effect on the life span of Kim. (In other words, assume that whether Andrew is alive in 20 years and whether Kim is alive in 20 years are independent events.) d) What is the probability that neither Andrew nor Kim lives for at least 20 years after Andrew retires?e) What is the probability that either Andrew or Kim (or both) live for at least 20 years after Andrew retires?
- Assume that a population of patients contains 30% of individuals who suffer from a certain fatal syndrome Z, which simultaneously makes it uncomfortable for them to take a life-prolonging drug X. Let Z = 1 and Z = 0 represent, respectively, the presence and absence of the syndrome, Y = 1 and Y = 0 represent death and survival, respectively, and X = 1 and X = 0 represent taking and not taking the drug. Assume that patients not carrying the syndrome, Z = 0, die with probability 0.5 if they take the drug and with probability 0.5 if they do not. Patients carrying the syndrome, Z = 1, on the other hand, die with probability 0.7 if they do not take the drug and with probability 0.3 if they do take the drug. Further, patients having the syndrome are more likely to avoid the drug, with probabilities p(X = 1|Z=0) = 0.9 and P(X = 1|Z = 1) = 0.6 . Based on this model, compute the joint distributions and for all values of x, y, and z. Present the following joint distributions in tables. [Hint:…A park ranger is searching for bears in a region of the park where on average there are 5 bears per square mile. The bears are solitary independent creatures, so it is reasonable to assume that the numbers of bears in disjoint regions are independent unknowns and that the number expected in any region is proportional to the area of the region. The ranger can also assume that in a very tiny region, say a square inch, it is impossible to find more than one bear. What is the probability (two decimal place accuracy) that he finds less than 12 bears in a given 2 square mile region of the park?As soon as possible! Let X and Y be continuous random variables with joint PDF