Unit 3 Problem Set (Cont.) 3. [20 pts] Recall that if X is a random variable wit
ID: 3353684 • Letter: U
Question
Unit 3 Problem Set (Cont.) 3. [20 pts] Recall that if X is a random variable with expected value E(X) SD(X) ,Var(X) a. What do Var(X) and SD(X) measure? b. Explain why Var(X) is guaranteed to be non-negative based on its definition, so SD(X) is well-defined Prove Var(X + c) answer to Part a c. Var(X) for any constant c. Interpret this result in light of your Prove Var(cX) = c2 Var(X) and SD (cX) = c SD (X) for any constant c. Interpret this result in light of your answer to Part a. d. e. Prove Var(x)-E(X2) - [E(x)]2. Why does this result also imply E(x) 2 E(X)]2 for any random variable X? Given any constant b, the nth moment of X about b is equal to E[(X - b)"]. Prove that for any n 1, the nth moment about b is minimized if b = E(X). Conclude that Var(X) is smaller than the 2nd moment of X about any other number. f.Explanation / Answer
A) Variance and standard deviation measures the spread of the distribution.V(X) = sum(Xi-Xbar)^2/(n-1) and sd(X) = SQRT(var X)
B) according to the formula of variance, V(X) = sum(Xi-Xbar)^2/(n-1). since the part (Xi-Xbar)^2 in the formula of variance is squared, it is not possible that the variance will be negative.
The standard deviation is defined as square root of variance. Since the variance is positive hence the standard deviation which is its square root is also positive.
C)
V(c+X) = E(c+X)^2-{E(c+X)}^2
=E(c^2 + X^2 + 2cX) - {c + E(X)}^2
= c^2 + E(X^2) + 2cE(X) - c^2 - {E(X)}^2 - 2c E(X)
= E(X^2) - {E(X)}^2
=V(X)
Sd(c+X) = sqrt V(c + X) = sqrt(V(X)) = sd(X)
As each observation is added by a constant number, there spread is the same. spread is measured as the sum of square of distances from mean. this sum of squares of distances is the same.
D)
V(aX) = E(aX)^2-{E(aX)}^2
=E(a^2 X^2)-{a E(X)}^2
=a^2E(X^2)-a^2 {E(X)}^2
=a^2 [ E(X^2)-{E(X)}^2 ]
=a^2 V(X)
Sd(cX) = sqrt V(cX) = sqrt(a^2 V(X)) = a sqrt (V(X)) = a SD(X)
As each observation is multiplied by a constant number, the new variance is square of constant of original variance.
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