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- The conditional probability of E given that F occurs is P(EF)=___________. So in rolling a die the conditional probability of the event E, getting a six, given that the event F, getting an even number, has occurred is P(EF)=___________.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?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?
- For a given electronic component, it has been shown that it is necessary to restart once every 250 hours. Let X be the number restarts within 1000 hours. (f) Given that the electronic component has been operating continuously for 200 hours, what is the probability that it will operate for another 400 hours?If X and Y are random variables and X is a geometric random variable where p = 0.1 then what is the probability mass function of Y = sin(X*pi) ?suppose that two radioactive particles are in a box. the expected time for each to decay is 5 hours. a)what is the expected time for both atoms to decay? b) what is the expected time for at least one of the two to decay?
- A 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?Amrita flips a biased coin repeatedly until she observes that both a head and a tail appear.The probability that a head will appear on a single flip is p, for 0 < p < 1. Find the probability mass function of N, the number of flips required to observe both a head and a tail. Hint: If N is the flip number that ends the experiment then that means that the N − 1 flips previous to flip N must all be the same as each other but different from the Nth flip. Also think about what the smallest value in the support of N must be. Finally remember that there are two cases: a sequence of tails followed by a head, or a sequence of head’s followed by a tail.If y is random variable has a probability mass function defined as follows find 1. and 2.
- 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.Suppose that the random variable X is continuous and takes its values uniformly over the interval from 0 to 2. What is P{X = 1.5 or X = 0.4}?If X is a continuous random variable with X ∼ Uniform([0, 2]), what is E[X^3]?
