The independent random variables X and Y are such that X ~ N(24, 25) and Y ~ N(40, 36). (a) Find (i) E(2X…

Question

The independent random variables X and Y are such that X ~ N(24, 25) and Y ~ N(40, 36). (a) Find (i) E(2X − 3Y), (ii) Var(2X − 3Y). (b) State the distribution that the random variable 2X − 3Y follows.

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a) i) Find E(2X − 3Y). Step 1: Use the linearity property of expectation, E(aX + bY) = aE(X) + bE(Y). E(2X − 3Y) = 2E(X) − 3E(Y); Step 2: Substitute the given values E(X) = 24 and E(Y) = 40. E(2X − 3Y) = 2(24) − 3(40); E(2X − 3Y) = 48 − 120; E(2X − 3Y) = −72 ii) Find Var(2X − 3Y). Step 1: Use the property for the variance of a linear combination of independent random variables, Var(aX + bY) = a²Var(X) + b²Var(Y). Var(2X − 3Y) = (2)²Var(X) + (−3)²Var(Y); Step 2: Substitute the given values Var(X) = 25 and Var(Y) = 36. Var(2X − 3Y) = 4(25) + 9(36); Var(2X − 3Y) = 100 + 324; Var(2X − 3Y) = 424 b) State the distribution that the random variable 2X − 3Y follows. Step 1: Since X and Y are independent normal random variables, any linear combination of X and Y is also a normal random variable. Step 2: The mean and variance of this new normal distribution are the values calculated in part (a). The distribution is N(μ, σ²), where μ = E(2X − 3Y) = −72 and σ² = Var(2X − 3Y) = 424. The random variable 2X − 3Y follows a normal distribution: N(−72, 424)

The independent random variables X and Y are such that X ~ N(24, 25) and Y ~ N(40, 36). (a) Find (i) E(2X - 3Y), (ii) Var(2X - 3Y). (b) State the distribution that the random variable 2X - 3Y follows.

Answers

Step-by-step solution

Part a(i)· 2 steps

Use the linearity property of expectation, $E(aX + bY) = aE(X) + bE(Y)$ .

Substitute the given values $E(X) = 24$ and $E(Y) = 40$ .

Part a(ii)· 2 steps424

Use the property for the variance of a linear combination of independent random variables, $Var(aX + bY) = a^2Var(X) + b^2Var(Y)$ .

Substitute the given values $Var(X) = 25$ and $Var(Y) = 36$ .

Part b· 2 stepsN(-72, 424)

Since X and Y are independent normal random variables, any linear combination of X and Y is also a normal random variable.

The mean and variance of this new normal distribution are the values calculated in part (a).

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