Handout Ch3 實習
微積分複習
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2
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 Chen
微積分笑話一
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3
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 Chen
微積分笑二
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4
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 Chen
微積分笑話三
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5
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 Chen
微積分笑話四
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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 Chen
Example 1
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Suppose that the p.d.f of a random variable X is as
follows：
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cx3 for1  x  2
f ( x)  
otherwise

0
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a. Find the value of constant c and sketch the p.d.f
b. Find the value of Pr(X>3/2)
Jia-Ying Chen
Solution
a.
2
4 3
x
15
4
f ' ( x)  x 2 when1  x  2, f' (x)  0
5
8
f ' ' ( x)  x when1  x  2, f' ' (x)  0
5
4
32
f( 1 ) 
f( 2 ) 
15
15
f ( x) 
 cx dx  1
3
1
1 4 2
cx 1  1
4
1
c(24  14 )  1
4
4
c
15
32
15
4
15
3
Pr( X  )
2
2
b.


4 3
x dx
15
3
2

8
1 4
x
15
2
3
2

35
48
Jia-Ying Chen
Cumulative Distribution Function
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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:
F ( x)  Pr(X  x)
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for    x  
Discrete distribution
F ( x)  Pr(X  x)   f (t )
tx
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Continuous distribution
x
F ( x)  Pr( X  x)   f (t )dt
9
f ( x)  F ( x) 
dF ( x)
dx
Jia-Ying Chen
Determining Probabilities from the c.d.f.
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For every x, Pr(X > x) = 1-F(x)
lim F ( x)  0
x  
lim F ( x)  1
x 
For all x1 and x2 such that x1 < x2, then
Pr(x1  X  x2 )  F ( x2 )  F ( x1 )
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For each x,
Pr(X  x)  F ( x  )
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 For every x, Pr(X  x)  F ( x)  F ( x )
For example, Pr(X  x1 )  z1  z0 ,
Pr(X  x3 )  z3  z 2
and the probability of every
other individual value of X is 0.
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Jia-Ying Chen
Example 2
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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 Chen
0.8
0.7
0.3
0.2
1
12
2
4
5
Jia-Ying Chen
Solution
a.Pr( X  2)  F (2)  F (2  )  0.3  0.3  0
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b.Pr(2  X  5)  F (5)  F (2 )  1  0.3  0.7
c.Pr( X  5)  1  F (5 )  1  1  0
d .Pr( X  4)  F (4)  F (4  )  0.8  0.7  0.1
e.Pr(1  X  2)  F (2)  F (1)  0.3  0.3  0
f .Pr(2  X  4)  F (4)  F (2  )  0.8  0.3  0.5
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Jia-Ying Chen
Bivariate Distributions
- Discrete Joint Distributions 
The joint probability mass function, or the joint p.m.f., of X and Y is
defined as
f ( x, y)  Pr(X  x and Y  y).
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Example: Suppose the joint p.m.f. of X and Y is specified as:
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
Pr(X  2 and Y  2)  f (2,2)  f (2,3)  f (2,4)  f (3,2)  f (3,3)  f (3,4)  0.5
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Pr(X  1)   f (1, y)  0.2
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y 1
Jia-Ying Chen
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,
Pr( x, y )  A   f ( x, y )dxdy
A
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The joint p.d.f. must satisfy two conditions:
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f ( x, y)  0 for    x   and    y  
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
  f ( x, y)dxdy  1
Jia-Ying Chen
雙重積分複習
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0≦y ≦1-x,0≦x≦1
1 1 y

f ( x, y )dxdy
y=1-x
0 0
1 1 x
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
0 0
積分秘訣
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f ( x, y )dydx
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
Jia-Ying Chen
Example 3
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Suppose that the joint p.d.f of two random
variables X and Y as follows：
5 2
2
(
x

y
)
for
0

y

1

x
,

f ( x, y )   4

0 otherwise
Determine Pr(0<=X<=1/2)
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Jia-Ying Chen
Solution
1
Pr(0  X  )
2
1
1 x 2 5
 2 
( x 2  y )dydx
0 0
4
1
5 2
5 2 1 x2
2
  ( x y  y ) 0 dx
0 4
8
1
5 2
5
2
2
  [ x (1  x )  (1  x 2 ) 2 ]dx
0 4
8
1
5 5 4
79
2
  (  x )dx 
0 8
8
256
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Jia-Ying Chen
Bivariate Cumulative Distribution
Functions
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The joint cumulative distribution function, or joint c.d.f., of two
random variables X and Y is defined as F ( x, y)  Pr(X  x and Y  y).
Note that Pr(a  X  b and c  Y  d )
 F (b, d )  F (a, d )  F (b, c)  F (a, c).
F1 ( x)  Pr( X  x)  lim F ( x, y )
y 
F2 ( y )  Pr(Y  y )  lim F ( x, y )
x 
y
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x
If X and Y have a continuous joint p.d.f., then F ( x, y)    f (r , s)drds
The joint p.d.f. can be derived from the joint c.d.f. by using
 2 F ( x, y )
f ( x, y ) 
xy
Jia-Ying Chen
Marginal Distributions
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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:
f1 ( x)  Pr(X  x)   Pr(X  x and Y  y )   f ( x, y ).
y
Also,
y
f 1 ( y )   f ( x, y )
x
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If X and Y have a joint p.d.f. f, then the marginal p.d.f. of X and Y are:

f1 ( x)    f ( x, y )dy for    x  

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f 2 ( y )    f ( x, y )dx for    y  
Jia-Ying Chen
Independent Random Variables
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Two random variables (discrete or continuous) X and Y are
independent if, for every two sets A and B of real numbers,
Pr(X  A and Y  B)  Pr(X  A) Pr(Y  B)
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Two random variables X and Y are independent if and only if,
for all real numbers x and y,
Pr(X  x and Y  y)  Pr(X  x) Pr(Y  y)
F ( x, y)  F1 ( x) F2 ( y)
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X and Y are independent if and only if, for all real numbers x
and y,
f ( x, y)  f1 ( x) f 2 ( y)
Jia-Ying Chen
Independent Random Variables
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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    x  
and    y  , f ( x, y)  g1 ( x) g 2 ( y)
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Proof:
f ( x, y )  g1 ( x ) g 2 ( y )
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
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Then 1    f ( x, y )dxdy   g1 ( x)dx  g 2 ( y )dy  C1C2


where C1   g1 ( x)dx and C2   g 2 ( y )dy. Also,

f1 ( x)   f ( x, y )dy  C2 g1 ( x),

f 2 ( y )   f ( x, y )dx  C1 g 2 ( y )
Since C1C2  1, we thus see that f ( x, y )  f1 ( x) f 2 ( y )
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Jia-Ying Chen
Example 4
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Suppose that X and Y have a discrete joint
distribution for which the joint p.f. is defined as
follow：
1
 ( x  y ), x  0,1y  0,1
f ( x, y )   4

0, otherwise
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a. Determine the marginal p.f.’s of X and Y
b. Are X and Y independent?
Jia-Ying Chen
Solution
1
1
b. f ( x, y )  ( x  y )  f ( x) f ( y )  (2 x  1)( 2 y  1)
4
16
24
Jia-Ying Chen
Discrete and Continuous
Conditional Distributions
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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
g1 ( x | y)  Pr(X  x | Y  y) 
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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
f ( x, y)
g1 ( x | y) 
f 2 ( y)
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Pr(X  x and Y  y) f ( x, y)

Pr(Y  y)
f 2 ( y)
for    x  
Jia-Ying Chen
Example 5 (3.6.7)
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Suppose that the joint p.d.f of two random variables
X and Y is as follows:
3
 (4  2 x  y)
f ( x, y )  16

0,
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for x  0, y  0 and 2 x  y  4
otherwise
Determine (a) the conditional p.d.f of Y for every
given value of X, and (b) Pr (Y  2 X  0.5)
Jia-Ying Chen
Solution
42 x
3
3
(4  2 x  y )dy  (8  8 x  2 x 2 )
0
16
16
 4  2x  y
for 0  y  4  2 x
f ( x, y ) 
So, g 2 ( y | x) 
 8  8x  2 x 2
f1 ( x)

ot herwise
 0
f1 ( x)  
Pr Y  2 | X  0.5   g 2  y | 0.5dy  
27
3
3
2
2
4  2x  y
1
dy

8  8x  2 x 2
9
Jia-Ying Chen