Worksheet 3: Applications in Probability, Assignments of Calculus

Download Worksheet 3: Applications in Probability and more Assignments Calculus in PDF only on Docsity! National Taiwan University - Calculus 2 for Class 06-12 Worksheet 3: Applications in Probability Name: ID: Department: Introduction. • Given a continuous random variable X whose domain is the whole real line, a probability density function (p.d.f.) of X is a non-negative function f(x) defined on R such that (i) Z 1 1 f(x) dx = 1, (ii) For any interval [a, b], the probability of X 2 [a, b] is given by P(X 2 [a, b]) = Z b a f(x) dx. • Given a probability density function f(x) of a continuous random variable X, we define its expected value E(X), its variance var(X) and its standard deviation (X) by E(X) = Z 1 1 xf(x) dx ; var(X) = Z 1 1 (x E(X))2f(x) dx ; (X) = p var(X). In particular, these definitions are defined by improper integrals. Exercise 1. Many random variables X are modelled by a normal distribution whose probability density function equals fX(x) = 1p 2⇡ e (xµ)2 22 , where µ, are constants and > 0. (a) It is known that Z 1 1 ex2 dx = p ⇡. By a suitable substitution, deduce that Z 1 1 1p 2⇡ e (xµ)2 22 dx = 1. (b) Compute the expected value and standard deviation of X. (c) Let f(x) = ce x2 8 + x 4 where c is a constant. Complete the square and rewrite f(x) in the form of 1p 2⇡ e (xµ)2 22 . Find c so that f(x) is a probability density function (i.e. Z 1 1 f(x) dx = 1) and find its expected value and standard deviation by just reading the simplified form of f(x). 1 恩 B 11505044 ⼯程科學⻢海洋⼯程學系 會發⽣的狀⼀ afisFFtezrxet " leea : nexe' ' tuizrsxuldxfeeatxoafixtdu… J⼆ 2就 fiiealeea 品 : u 球 ☆ wewdw? _ 抗×π= 1 ⽢ 是wdw = du ElJxtxxdx = xukIfxtufxx) =fxifxcx)tuftxs.∴x - x = J ☆ xl (* 1 x) + n =f。x☆ e"tu= zh 地 zriicxMx 、 o鼠 e - " + M = O + M = U varlx) : x-ulpfaie' " =f x - 0 (x-u 1[ -xue -Ʃ" } dx x J 2π Intigration bypartfzatx-u) -fen *m " e -rerxul50 | . - e x 七虎 dx letJµ= w IF 2 ☆ 1 = O - TIafoPe - Enlxupeeui " … inexidx * Je “xixxrr -nfetix 2wd2 U = w 之 du= 2wdw National Taiwan University - Calculus 2 for Class 06-12 Worksheet 3: Applications in Probability Name: ID: Department: Introduction. • Given a continuous random variable X whose domain is the whole real line, a probability density function (p.d.f.) of X is a non-negative function f(x) defined on R such that (i) Z 1 1 f(x) dx = 1, (ii) For any interval [a, b], the probability of X 2 [a, b] is given by P(X 2 [a, b]) = Z b a f(x) dx. • Given a probability density function f(x) of a continuous random variable X, we define its expected value E(X), its variance var(X) and its standard deviation (X) by E(X) = Z 1 1 xf(x) dx ; var(X) = Z 1 1 (x E(X))2f(x) dx ; (X) = p var(X). In particular, these definitions are defined by improper integrals. Exercise 1. Many random variables X are modelled by a normal distribution whose probability density function equals fX(x) = 1p 2⇡ e (xµ)2 22 , where µ, are constants and > 0. (a) It is known that Z 1 1 ex2 dx = p ⇡. By a suitable substitution, deduce that Z 1 1 1p 2⇡ e (xµ)2 22 dx = 1. (b) Compute the expected value and standard deviation of X. (c) Let f(x) = ce x2 8 + x 4 where c is a constant. Complete the square and rewrite f(x) in the form of 1p 2⇡ e (xµ)2 22 . Find c so that f(x) is a probability density function (i.e. Z 1 1 f(x) dx = 1) and find its expected value and standard deviation by just reading the simplified form of f(x). 1 ( x- 1 ) 2 - 旦 x 2 ⽉ 1 《 ) f(x ) = Ce - '+☆= CCSxtz ' = C e * ☆ e 5 = Cxeixe =⇒ cxe 式 = xπ xz =⇒ c= 2 u = 1 X Exercise 4. Suppose X is a continuous random variable with standard normal distribution and set Y = X2. (a) Write down the distribution function of Y as an integral. (You don’t need to evaluate it.) (b) Find the probability density function of Y . This probability density function is called the Chi-squared distribution. (c) Compute the expected value and variance of Y . 4 @ 1 f (x ) = 2π e的 F( VH ( x ≤ x) = fxxx)d Yco + F(y ) = 0 — * Fy 1 = PCY ≤ y ) = ptxEy ) =pt -E " -, " flx) dx * 此 Y ≤01F (y ) = O * F(y ) =f( y )=f(MN -ft~⼀ □ "xe = x 2 *房 =e⾳ g 《 1 ∞ " xydy 。 Jo⼀就代 。 少 = 就 ( 刌 ⼀11"ze - *10 % 0 “就 (ye * 10 tf ) ⼆→ e -i ⼀ = O ×O = 0 Y-→x ixe三 lin - e ”:⾶。 “我之 )…d 、“ … …⼆ 1 2udu = idy Varl )在下⼀⾴ Exercise 4. Suppose X is a continuous random variable with standard normal distribution and set Y = X2. (a) Write down the distribution function of Y as an integral. (You don’t need to evaluate it.) (b) Find the probability density function of Y . This probability density function is called the Chi-squared distribution. (c) Compute the expected value and variance of Y . 4 c 1fy -1Pe三 f .ty 州eidy =9↑ y^5 e Ʃ+ 9-zye - ↓+ f 。 e"dy :9 . 0 ye " y -zxl + = f% ∞⼭……⼭ integratinn by part Jyye 三⼩ y = ( )Jze 三 xxyyJyxze⼀ )⼀ 。 1 」 (Yyze * 18 ) +3(59 . "e= J2π ⼆ 0 3 -15 = 2 * 3 zxy 性 _ li州老于 yi = 0筋e玩怀 xiez