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Calculating the Mean and Variance of a Poisson Random Variable

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1. Let X be a random variable with probability density function given by

f(x) - 2(1 - x), 0 <= x <= 1,
0, otherwise

a. Find the density function of Y - X^2
b. Find the mean and variance of Y.

2. Let X be a random variable with probability density distribution given by

f(x) - x, 0 <= x <= 1
1 < x <= 1.5
0, otherwise

Find the probability density function of Y - 10X - 4

3. Calculate the mean and variance of a Poisson random variable with parameter lambda.

4. Calculate the mean and variance of a Gamma random variable with parameter alpha and lambda.

5. Show that the characteristic function of the Cauchy random variable is c^-|w|.

Find the probability density function of Y - 10X - 4.

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