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Handout Ch3 實習

Handout Ch3 實習. 微積分複習. dx n /dx=nx n-1 dC/dx=0 dlnx/dx=1/x de x /dx=e x dx/dy=0 ∫x n dx=(1/n+1)x n+1 +C ∫e x dx=e x +C ∫(1/x)dx=ln∣x ∣+C. 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。」

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Handout Ch3 實習

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  1. Handout Ch3 實習

  2. 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

  3. 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！」

  4. 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…

  5. 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。」

  6. 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！」

  7. 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)

  8. Solution a. b.

  9. 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

  10. 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.

  11. 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)

  12. 0.8 0.7 0.3 0.2 5 4 1 2

  13. Solution

  14. 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:

  15. 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:

  16. y=1-x 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

  17. Example 3 • Suppose that the joint p.d.f of two random variables X and Y as follows： Determine Pr(0<=X<=1/2)

  18. Solution

  19. 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

  20. Marginal Distributions • 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:

  21. Independent Random Variables • Two random variables (discrete or continuous) X and Y areindependentif, 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,

  22. 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:

  23. 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?

  24. Solution

  25. 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

  26. 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)

  27. Solution

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