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EL630 HW4

# EL630 HW4 - EL630 Homework 4 1(Book 4-9 Find f x if F x...

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1 EL630: Homework 4 1. (Book: 4-9) Find ( ) if ( ) (1 ) ( ). x f x F x e u x c α = (Hint: For any function ( ), h x we have ( ) ( ) ( ) ( ), h x x a h a x a δ δ = which is called the sampling property of the impulse function.) 2. (Book: 4-10) If is (0,4) find (a) {1 2} and (b) {1 2| 1}. N P P X X X X 1 2 c x ( ) F x

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2 (Hint: (0,4) N means 2 2 ( ) / 2 2 1 ( ) 2 x f x e µ σ πσ = X with 2 0, 2 µ σ = . Then we have {1 2} (2) (1) P F F = X X X . 2 2 ( ) / 2 2 1 ( ) ( ) ( ) 1/ 2 2 x y x x F x e dy G erf µ σ µ µ σ σ πσ −∞ = = = X . Then use Table 4-1.) 3. (Book: 4-18) Show that ( ) ( | ) ( ) ( | )[1 ( )]. P A P A x F x P A x F x = + > X X 4. (Book: 4-19) Show that ( | ) ( ) ( | ) . ( ) x x P A x F x F x A P A = X 5. (Book: 4-27) A coin is tossed an infinite number of times. Show that the probability that k heads are observed at the n th toss but not earlier equals 1 . 1 n k n k p q k [See also book (4-63).] 6. (Book: 4-30) The probability that a driver will have an accident in 1 month equals 0.02. Find the probability that in 100 months he will have three accidents.
3 7. (Book: 4-31) A fair die is rolled five times. Find the probability that “one” shows twice,

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