Chapter 6: Q.6.1 (page 275)

Verify Equation $(1.2)$ . $P (a_{1} < X \leq a_{2}, b_{1} < Y \leq b_{2}) = F (a_{2}, b_{2}) + F (a_{1}, b_{1}) - F (a_{1}, b_{2}) - F (a_{2}, b_{1})$

Short Answer

Expert verified

Equation is proved.

$P (a_{1} < X \leq a_{2}, b_{1} < Y \leq b_{2}) = F (a_{2}, b_{2}) + F (a_{1}, b_{1}) - F (a_{1}, b_{2}) - F (a_{2}, b_{1})$

Step by step solution

Equation :

A declaration (represented by the symbol =) that the values of two mathematical expressions are equal.

Explanation :

Let us define the events,

$A : \{X \leq a_{1}\} B : \{X \leq a_{2}\} C : \{Y \leq b_{1}\} D : \{Y \leq b_{2}\}$

$P [a_{1} < X < a_{2} \cap b_{1} \leq Y \leq b_{2}] = P [(B - A) \cap (D - C)] = P [B \cap (D - C) - A \cap (D - C)] . . . . . . (1) [B y d i s t r i b u t i v e l a w]$

We know that if $E \subset F \Rightarrow E \cap F = E$ , then

$P (F - E) = P (\bar{E} \cap F) = P (F) - P (E \cap F) = P (F) - P (E) . . . . . . . (2)$

Obviously $A \subset B \Rightarrow [A \cap (D - C)] \subset [B \cap (D - C)]$

Hence by using $(2)$ , we get from $(1)$

$P [a_{1} < X \leq a_{2} \cap b_{1} < Y \leq b_{2}] = P [B \cap (D - C)] - P [A \cap (D - C)] = P [(B \cap D) - (B \cap C)] - P [(A \cap D) - (A \cap C)] = P (B \cap D) - (B \cap C) - P (A \cap D) + P (A \cap C) . . . . . . (3)$

We have localid="1647341297556" $P (B \cap D) = P [X \leq a_{2} \cap Y \leq b_{2}] = F (a_{2}, b_{2})$

Similarly localid="1647341325673" $P (B \cap C) = F (a_{2}, b_{1}) P (A \cap D) = F (a_{1}, b_{2})$

Substitute these values in $(3)$ we get

$P (a_{1} < X \leq a_{2}, b_{1} < Y \leq b_{2}) = F (a_{2}, b_{2}) + F (a_{1}, b_{1}) - F (a_{1}, b_{2}) - F (a_{2}, b_{1})$

Hence proved.

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Verify Equation $(1.2)$ . $P (a_{1} < X \leq a_{2}, b_{1} < Y \leq b_{2}) = F (a_{2}, b_{2}) + F (a_{1}, b_{1}) - F (a_{1}, b_{2}) - F (a_{2}, b_{1})$

Short Answer

Step by step solution

Equation :

Explanation :

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Verify Equation 1.2.P(a1<X≤a2,b1<Y≤b2)=F(a2,b2)+F(a1,b1)−F(a1,b2)−F(a2,b1)

Short Answer

Step by step solution

Equation :

Explanation :

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Most popular questions from this chapter

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Verify Equation $(1.2)$ . $P (a_{1} < X \leq a_{2}, b_{1} < Y \leq b_{2}) = F (a_{2}, b_{2}) + F (a_{1}, b_{1}) - F (a_{1}, b_{2}) - F (a_{2}, b_{1})$