Find the following probabilities for two normal random variablesZ=N(0,1) andX=N(−1,9). (a)P(Z >−1.48). (b)P(|X|<2.30). (c) What is...

Find the following probabilities for two normal random variablesZ=N(0,1) andX=N(−1,9).

(a)P(Z >−1.48).

(b)P(|X|<2.30).

(c) What is the type and the parameters of the random variableY= 3X+ 5?

Expert Solution

(a) P(Z > - 1.48) = 1 - P(Z - 1.48) = 1 - (-1.48) = 1 - 0.0694 = 0.9306

[(.) is the cdf of N(0,1)]

(b) X ~ N(-1, 9) i.e. (X + 1)/3 ~ N(0,1)

P(|X| 2.30) = P-2.30 X 2.30) = P[(-2.30 + 1)/3 (X + 1)/3 (2.30 + 1)/3] = P[-0.4333 (X + 1)/3 1.1] = P[(X + 1)/3 1.1] - P[(X + 1)/3 - 0.4333] = (1.1) - (-0.4333) = 0.8643 - 0.3324 = 0.5319

(c) We have, E(Y) = 3 * E(X) + 5 = -3 + 5 = 2

V(Y) = * V(X) = 9 * V(X) = 9 * 9 = 81

Since, Y is a linear function of X, Y ~ N(2, 81)

orchestra answered 2 years ago

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