6. Designing random variables and events to have certain properties. Part (a) and (b) are separate. Let X be a random variable. Is it always true that E[1/X] If so, then show why it's true. Otherwise find a counterexample to show why it is not (а) 1/E[X] ? %3D true. (b) Suppose we toss a biased (with q = P(Heads) E (0, 1)) coin 3 times and %3D let E-ΗΗΗ, HΤH, THH, TTH) %3D Find an example of one event F whose probability is neither zero nor one and which is also independent of E. Justify your answer.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![6.
Designing random variables and events to have certain properties. Part (a) and
(b) are separate.
(a)
Let X be a random variable. Is it always true that E[1/X] = 1/E[X] ?
If so, then show why it's true. Otherwise find a counterexample to show why it is not
true.
(b)
let
Suppose we toss a biased (with q P(Heads) E (0, 1)) coin 3 times and
E =
{HHH, HTH,THH,TTH}
Find an example of one event F whose probability is neither zero nor one and which is
also independent of E. Justify your answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F93c32466-9203-4d9d-9a0d-623c28291a42%2F521dfb34-1eae-46ee-a56a-9f46764cd3c6%2F2koph1_processed.jpeg&w=3840&q=75)
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