Let A, B, and C be independent random variables, uniformly distributed on (0, 1). What is the probability that Ax² +Ba+C has real roots? (Hint: If X~ uniform(0, 1), then - log X exponential. The sum of two independent exponentials is gamma.)
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A: Solution: From the given information,
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- 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 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?A bin of 5 transistors is known to contain 2 that are defective. The transistors are to be tested, one at a time, until the defective ones are identified. Denote by N1 the number of tests made until the first defective is identified and by N2 the number of additional tests until the second defective is identified. Find the joint probability mass func- tion of N1 and N2.
- Suppose that the time (in hours) required to repair a machine is an exponentially distributed random variable with parameter λ=0.9. What is(a) the probability that a repair time exceeds 9 hours? (b) the conditional probability that a repair takes at least 10 hours, given that it takes more than 7 hours?What is the probability that a SSRW on Z^2 starting at (0, 0) eventually hits the point (100, 100)?According to some law in genetics, when certain types of peas are crossed, the probability that the plant yields a yellow pea is 0.75 and that it yields a green pea is 0.25. For a plant yielding 90 peas, find the approximate probability that at most 64 peas are yellow. Use the normal approximation with continuity correction.
- If X, Y are normally distributed independent random variables, then show that W = 2X - Y is normally distributed. Do not use moment generating function, only use convolution formula.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?