Chapter 8: Q.8.3 (page 393)

If $E [X] = 75 E [Y] = 75 Var (X) = 10$
$Var (Y) = 12 Cov (X, Y) = - 3$
give an upper bound to
(a) $P {| X - Y | > 15};$
(b) $P {X > Y + 15};$
(c) $P {Y > X + 15}$ ;

Short Answer

Expert verified

An upper bound to

(a). $P {| X - Y | \geq 15} \leq . 1244$

(b). $P {X > Y + 15} \leq . 1107$

(c). $P {Y > X + 15} \leq . 1107$

Step by step solution

Part (a) Step 1: Given Information

Let $X$ and $Y$ be radnom variables such that

E [X] = 75, E [Y] = 75, Var (X) = 10

Var (Y) = 12, Cov (X, Y) = - 3

Part (a) Step 2: Explanation

Consider the random variable

$A = X - Y$

Since

$μ_{A} = E [A] = E (X) - E (Y) = 0$

and

$σ_{A}^{2} = Var (A) = Var (X) + Var (Y) - 2 Cov (X, Y) = 28$

using Chebyshev's inequality we get

$P {| X - Y | \geq 15} = P \{|A - μ_{A}| \geq 15\} \leq \frac{σ_{A}^{2}}{15^{2}} = \frac{28}{15^{2}} = . 1244$

Part (b) Step 1: Given Information

Let $X$ and $Y$ be random variables such that

localid="1650024398003" $E [X] = 75, E [Y] = 75, Var (X) = 10 Var (Y) = 12, Cov (X, Y) = - 3$

Part (b) Step 2: Explanation

Again, consider the random variable $A$ . Now, using Proposition 5.l we get

$P {X > Y + 15} = P {A > 15} \leq \frac{σ_{A}^{2}}{σ_{A}^{2} + 15^{2}} = \frac{28}{28 + 15^{2}} = . 1107$

Part (c) Step 1: Given Information

Let $X$ and $Y$ be radnom variables such that

E [X] = 75, E [Y] = 75, Var (X) = 10

Var (Y) = 12, Cov (X, Y) = - 3

Part (c) Step 8: Explanation

If we denote as $B = Y - X$ , then this random variable has mean

$μ_{B} = E [B] = E (Y) - E (X) = 0$

and variance

$σ_{B}^{2} = Var (B) = Var (Y) + Var (X) - 2 Cov (X, Y) = 28$

Using One-sided Chebyshev inequality we get

$P {Y > X + 15} = P {B > 15} \leq \frac{σ_{B}^{2}}{σ_{B}^{2} + 15^{2}} = \frac{28}{28 + 15^{2}} = . 1107 .$

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If $E [X] = 75 E [Y] = 75 Var (X) = 10$
$Var (Y) = 12 Cov (X, Y) = - 3$
give an upper bound to
(a) $P {| X - Y | > 15};$
(b) $P {X > Y + 15};$
(c) $P {Y > X + 15}$ ;

Short Answer

Step by step solution

Part (a) Step 1: Given Information

Part (a) Step 2: Explanation

Part (b) Step 1: Given Information

Part (b) Step 2: Explanation

Part (c) Step 1: Given Information

Part (c) Step 8: Explanation

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If E[X]=75E[Y]=75Var(X)=10Var(Y)=12Cov(X,Y)=-3give an upper bound to(a) P{|X-Y|>15};(b) P{X>Y+15};(c)P{Y>X+15};

Short Answer

Step by step solution

Part (a) Step 1: Given Information

Part (a) Step 2: Explanation

Part (b) Step 1: Given Information

Part (b) Step 2: Explanation

Part (c) Step 1: Given Information

Part (c) Step 8: Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Statistics

Mechanics Maths

Applied Mathematics

Logic and Functions

Decision Maths

Study anywhere. Anytime. Across all devices.

If $E [X] = 75 E [Y] = 75 Var (X) = 10$
$Var (Y) = 12 Cov (X, Y) = - 3$
give an upper bound to
(a) $P {| X - Y | > 15};$
(b) $P {X > Y + 15};$
(c) $P {Y > X + 15}$ ;