c d e Let Z~ Normal(0, 1).(a) Find the "absolute first moment" E[[Z].(b) Use Markov's inequality and Chebyshev's inequality to estimate the probabilityP(|Z| > 2). Compute also the actual value of the probability (using Z-table). Comparethe result.(c) Let Y~ Normal(µ, σ²), and consider the probability P(|Y − µ| > ro) for some r > 0.Show that the probability is independent of u, o (with o > 0).(d) Let X = eºZ+µ (o> 0, µ € R). Compute the density of X and E[X].(e) A chi-square random variable, denoted Q ~ x²(k) for k e N (k is called the degrees offreedom) is defined by Q = E₁ Z², where Zi's are independent standard normal distribution.Compute the expected value of Q.

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A First Course in Probability (10th Edition)

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(c) Let Y~ Normal (μ, o²), and consider the probability P(|Y - µ| > ro) for some r > 0. Show that the probability is independent of u, o (with o > 0).