Fill in blanks. If Var(2X + 1) = 4 then Var(l - 3X) = _________ F(a) is the dist

Question

Fill in blanks. If Var(2X + 1) = 4 then Var(l - 3X) = _________ F(a) is the distribution function of a random variable X. The sentence that " X has probability 0.67 of being less than or equal to 2. S' can be expressed in terms of F as _________. X is a random variable which can take on value of 1 with probability .3 and value of 2 with probability .7. The expectation value of X^2 is then E[X^2] = _________. Suppose E[X] = 5 and E[X(X - 1)] = 30, then Var(X) = _________. If E[X] = -1 and Var(X) = 2, then E[3X - 10] = _________. If E[X] = -1 and Var(X) = 2, then Var(3X - 10) = _________.

Explanation / Answer

a) Var(2X + 1) = 4

or, 22 * Var(X) = 4

or, Var(X) = 1

Var(1 - 3X) = (-3)2 * Var(X)

                   = 9 * 1

                   = 9

b) F(2.5) = 0.67

c) E(X2) = 12 * 0.3 + 22 * 0.7 = 3.1

d) E(X) = 5

E[X(X - 1)] = 30

or, E(X2) - E(X) = 30

or, E(X2) - 5 = 30

or, E(X2) = 35

Var(X) = E(X2) - (E(X))2

           = 35 - 52

           = 10

e) E(3X - 10) = 3 * E(X) - 10

                     = 3 * (-1) - 10

                     = -13

f) Var(3X - 10) = 32 * Var(X)

                       = 9 * 2

                       = 18

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