Download - Handout Ch3 實習
Handout Ch3 實習
Handout Ch3 實習
Jia-Ying Chen2
微積分複習
dxn/dx=nxn-1
dC/dx=0 dlnx/dx=1/x dex/dx=ex
dx/dy=0 ∫xndx=(1/n+1)xn+1+C ∫exdx=ex+C ∫(1/x)dx=ln x +C∣ ∣
Jia-Ying Chen3
微積分笑話一 某天,一位同學和微積分教授說: 「教授啊,我今天心情很不好耶…」 教授就說:「那我用微積分來幫你卜卦看看好不好?」 於是,教授就要求同學隨意寫下兩個字,同學雖然半信半疑,但是還是寫了「麻煩」二字。
教授看了之後,笑笑的說:「你一定是被爸媽罵了。」 同學大驚:「哇塞!教授,你怎麼那麼厲害,一下就猜到了!」 「你別急,我來慢慢解釋給你聽。」教授不急不徐地解釋: 「首先我們先用一次微分把麻煩的「麻」字的蓋子微掉, 不就剩下「林」了嗎?然後也把「煩」這個字用二次微分, 分別把「火」和「┬」去掉,剩下的字就是「貝」。」
「此時我們可以得到「林貝」二字,這就說明你被你爸罵了!」 正當同學張大嘴巴說不出話來時,教授又繼續說了下去。 「還沒完喔,現在再把剩下的「貝」字再做一次微分, 把下面的「八」去掉,就得到「目」這個字。」
「因此我們又得到「林目」二字,這說明你也有被你媽媽罵!」
Jia-Ying Chen4
微積分笑二
某天上微積分課時,教授提到了在座標軸上的積分,學生們看著滿滿的黑板公式,趕緊在下面抄筆記,但是心似乎都不放在課堂之上。
講到一半,教授問一位女同學:「先積甚麼?」 女同學被突如其來的問話嚇了一跳,緊接著說她不會,教授再問全班同學,也沒有人回答。
這時教授突然大吼一聲:「雞歪啦!連這個都不會。」 全班同學當場嚇了一大跳,教授竟然開口飆髒話! 結果仔細一看,才發現教授正在積y軸…
Jia-Ying Chen5
微積分笑話三 有一位數學家得了精神病,他覺得自己是微分的主宰者,朋友們將他送到精神病院希望他能好起來。
每天,這位數學家都會在院內四處閒逛,只要遇到其他病人,他就會恐嚇地說:「我要把你微分掉!」
有一天,院裡來了一個新病人,像往常一樣地,他瞪著那位病人大聲吼:「我要把你微分掉!」但這次,那位病人的表情一點也不變。
數學家十分訝異,提起精神來狠狠地盯著那位新病人,更大聲地說:「我要把你微分掉!」但那位病人依然一點反應也沒有。
數學家氣極了,最後他放聲大叫:「我要把你微分掉!」 病人平靜地看了數學家一眼,他說: 「你想怎麼微分我都無所謂,因為我是e的x次方。」
Jia-Ying Chen6
微積分笑話四
某天,常數函數C和指數函數e的x次方走在街上,遠遠地,他們看到微分運算元朝他們這邊走了過來。
常數函數嚇得慌忙躲藏起來,緊張地說:「被它微分一下,我就什麼都沒有啦!」
指數函數則是不慌不忙地說:「它可不能把我怎麼樣,我可是e的x次方耶!」
終於,指數函數和微分運算元在路中相遇了。 指數函數首先自我介紹道:「你好,我是e的x次方!」 而微分運算元也微笑地自我介紹: 「你好,我是d/dy!」
Jia-Ying Chen7
Example 1
Suppose that the p.d.f of a random variable X is as follows :
a. Find the value of constant c and sketch the p.d.f b. Find the value of Pr(X>3/2)
3 1 2( )
0
cx for xf x
otherwise
Jia-Ying Chen8
Solutiona.
b.
23
1
4 21
4 4
1
11
41
(2 1 ) 14
4
15
cx dx
cx
c
c
15
322
15
41
0(x)'f'2,x1 when 5
8)(''
0(x)f'2,x1 when 5
4)('
15
4)(
2
3
) f()f(
xxf
xxf
xxf
15
4
32
15
23
3
2
4 23
2
3Pr( )
2
4
15
1 35
15 48
X
x dx
x
Jia-Ying Chen9
Cumulative Distribution Function
The cumulative distribution function (c.d.f.) or distribution function (d.f.) of a random variable X (discrete or continuous) is a function defined for each real number x as follow:
Discrete distribution
Continuous distribution
xxXxF for )Pr()(
xt
tfxXxF )()Pr()(
dx
xdFxFxfdttfxXxF
x )()()( )()Pr()(
Jia-Ying Chen10
Determining Probabilities from the c.d.f.
For every x, Pr(X > x) = 1-F(x) For all x1 and x2 such that x1 < x2, then
For each x,
For every x,
For example,
and the probability of every
other individual value of X is 0.
0)(lim
xFx
)()()Pr( 1221 xFxFxXx
)()Pr( xFxX
)()()Pr( xFxFxX
,)Pr( 011 zzxX
233 )Pr( zzxX
1)(lim
xFx
Jia-Ying Chen11
Example 2
Suppose that the d.f. F of a random variable X is as sketched as follows. Find each of the following probabilities
a. Pr(X=2)
b. Pr(2<=x<=5)
c. Pr(X>=5)
d. Pr(X=4)
e. Pr(1<x<=2)
f. Pr(2<=X<=4)
Jia-Ying Chen12
1 2 4 5
0.2
0.3
0.7
0.8
Jia-Ying Chen13
Solution
.Pr( 2) (2) (2 ) 0.3 0.3 0
.Pr(2 5) (5) (2 ) 1 0.3 0.7
.Pr( 5) 1 (5 ) 1 1 0
.Pr( 4) (4) (4 ) 0.8 0.7 0.1
.Pr(1 2) (2) (1) 0.3 0.3 0
.Pr(2 4) (4) (2 ) 0.8 0.3 0.5
a X F F
b X F F
c X F
d X F F
e X F F
f X F F
Jia-Ying Chen14
Bivariate Distributions - Discrete Joint Distributions -
The joint probability mass function, or the joint p.m.f., of X and Y is defined as
Example: Suppose the joint p.m.f. of X and Y is specified as:
.) and Pr(),( yYxXyxf
Y
X 1 2 3 4
1 0.1 0 0.1 0
2 0.3 0 0.1 0.2
3 0 0.2 0 0
4
1
2.0),1()1Pr(
5.0)4,3()3,3()2,3()4,2()3,2()2,2()2 and 2Pr(
y
yfX
ffffffYX
Jia-Ying Chen15
Bivariate Distributions- Continuous Joint Distributions -
The joint probability density function, or the joint p.d.f. of X and Y is defined as f (x, y). For every subset A of the xy-plane,
The joint p.d.f. must satisfy two conditions:
A
dxdyyxfAyx ),(),(Pr
yxyxf and for 0),(
1),( dxdyyxf
Jia-Ying Chen16
雙重積分複習
0 y 1-x,0 x 1≦ ≦ ≦ ≦
積分秘訣
依照題目給定範圍畫出圖 判斷先積 x還是先積 y比較容易
11
0 0
( , )y
f x y dxdy
1 1
0 0
( , )x
f x y dydx
y=1-x
Jia-Ying Chen17
Example 3
Suppose that the joint p.d.f of two random variables X and Y as follows:
Determine Pr(0<=X<=1/2)
otherwise
xyforyxyxf
0
,10)(4
5),(
22
Jia-Ying Chen18
Solution
2
2
11 22
0 0
12 2 12
00
12 2 2 22
0
142
0
1Pr(0 )
2
5( )
4
5 5( )4 8
5 5[ (1 ) (1 ) ]4 8
5 5 79( )8 8 256
x
x
X
x y dydx
x y y dx
x x x dx
x dx
Jia-Ying Chen19
Bivariate Cumulative Distribution Functions The joint cumulative distribution function, or joint c.d.f., of two
random variables X and Y is defined as
Note that
If X and Y have a continuous joint p.d.f., then The joint p.d.f. can be derived from the joint c.d.f. by using
). and Pr(),( yYxXyxF
).,(),(),(),(
) and Pr(
caFcbFdaFdbF
dYcbXa
),(lim)Pr()(
),(lim)Pr()(
2
1
yxFyYyF
yxFxXxF
x
y
y x
drdssrfyxF ),(),(
yx
yxFyxf
),(
),(2
Jia-Ying Chen20
If X and Y have a discrete joint distribution for which the joint p.m.f. is f, then the marginal p.m.f. f1 of X can be found as follows:
Also,
If X and Y have a joint p.d.f. f, then the marginal p.d.f. of X and Y are:
Marginal Distributions
.),() and Pr()Pr()(1 y y
yxfyYxXxXxf
x
yxfyf ),()(1
ydxyxfyf
xdyyxfxf
for ),()(
for ),()(
2
1
Jia-Ying Chen21
Independent Random Variables Two random variables (discrete or continuous) X and Y are
independent if, for every two sets A and B of real numbers,
Two random variables X and Y are independent if and only if, for all real numbers x and y,
X and Y are independent if and only if, for all real numbers x and y,
)Pr()Pr() and Pr( BYAXBYAX
)()(),(
)Pr()Pr() and Pr(
21 yFxFyxF
yYxXyYxX
)()(),( 21 yfxfyxf
Jia-Ying Chen22
Independent Random Variables Suppose X and Y are random variables that have a continuous joint
p.d.f. Then X and Y will be independent if and only if, for and
Proof:
x)()(),( , 21 ygxgyxfy
)()(),( that see thus we,1 Since
)(),()( ),(),()(
Also, .)( and )( where
)()(),(1Then
)()(),(
2121
212121
2211
2121
21
yfxfyxfCC
ygCdxyxfyfxgCdyyxfxf
dyygCdxxgC
CCdyygdxxgdxdyyxf
ygxgyxf
Jia-Ying Chen23
Example 4
Suppose that X and Y have a discrete joint distribution for which the joint p.f. is defined as follow:
a. Determine the marginal p.f.’s of X and Y b. Are X and Y independent?
otherwise
yxyxyxf
,0
1,01,0),(4
1),(
Jia-Ying Chen24
Solution
)12)(12(16
1)()()(
4
1),(. yxyfxfyxyxfb
Jia-Ying Chen25
Discrete and Continuous Conditional Distributions
Suppose that X and Y have a joint p.m.f. f (x, y), then we can define the conditional p.m.f. g1 of x given that Y = y as
Suppose that X and Y have a joint p.d.f. f(x,y), then we can define the conditional p.d.f. g1 of X given that Y=y as
)(
),(
)Pr(
) and Pr()|Pr()|(
21 yf
yxf
yY
yYxXyYxXyxg
12
( , )( | ) for
( )
f x yg x y x
f y
Jia-Ying Chen26
Example 5 (3.6.7)
Suppose that the joint p.d.f of two random variables X and Y is as follows:
Determine (a) the conditional p.d.f of Y for every given value of X, and (b)
3(4 2 ) for 0, 0 and 2 4
( , ) 160, otherwise
x y x y x yf x y
)5.02(P XYr
Jia-Ying Chen27
Solution
otherwise 0
24y0for 288
24
)(
),()|( So,
)288(16
3)24(
16
3)(
2
12
24
0
21
xxx
yx
xf
yxfxyg
xxdyyxxfx
3
2
3
2 22 9
1
288
245.0|5.0|2Pr dy
xx
yxdyygXY