Given a random variable Y ~ Pois(A) (a Poisson random variable with parameter ) > 0), (a) Find My (t) (b) Use this result to get the mean and variance of Y.

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A: Given that fy=30y21-y2, 0&lt;y&lt;1 Formula for Ey4 is, Ey4=∫01y4fydy

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Q: If X is a Gaussian random variable with mean zero and variance o, find the pdf of Y = \XI.

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Q: Let X Poisson(A) and Y = vX. Use the delta method to approximate the mean and variance of Y.

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Q: If has a Poisson distribution such that P(X 21)=0.8646 then then the variance of is equal to

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A: The answer is given using the concept of the Bernoulli random variable.

Q: 6. Show that 1 s2 = TE2-1(Xi – x)² is unbiased estimator of the population variance o?

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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!