第一胎生男生的机率p

第一胎生男生的机率p Lecture 05. Probability Distributions ?A distribution is a mathematical function used to describe (model, modeling) a set of observed data. ?A random variable is a function defined on a set of events, and takes real value. Example 1. For tossing a dice one time, let A={odd}, and B={even}. And if you get an odd at one tossing, you win 1 dollor; otherwise you loss 1 dollor. It is of course a fair game if the dice is „fair?! That means, if we denote X as a random variable representing the money that you win or loss, then X(A)=1, and X(B)=,1. [Note cthat, in this case, B=A, which implies A?B=Ω.] The function X thus defined is called a discrete random variable; it only takes values 1 and ,1. Moreover, The „probability distribution? of X is: P(X=1)=0.5 and P(X=-1)=0.5, because we have just claimed that the dice is fair. 5 Example 2. Let F(,) denote a population with size 10. It comprises people who are older than 50 years of age. If we denote w to be a sample point representing an sampled individual, and X(w) is her/his measured systolic blood pressure. X(,) is suitably to be viewed as a continuous random variable, and the probability distribution of X is a continuous function, say, f(x), which satisfies the following properties: (1) f(x)>=0, for -?0, x is in R}, then ?f(x)dx=1, S x and F(x)=?f(u)du. As stated in a previous lecture, f(x) is called the probability density function (pdf), and F(x) the cumulative distribution function (cdf) with dF(x)/dx=f(x). The Bernoulli distribution Ber(p) Formulae Tossing a coin, Pr(Head)=p, Pr(Tail)=1-p Random Variable X DEF. : If „Head?, then “+1”; if „Tail?, then “+0”. We have P(X=1)=p; P(X=0)=1-p; 035歲之婦女,其終生得乳癌之 機率=p, 不會得乳癌之機率=1- p 11 3. 生第一胎的<20歲之婦女,其終生得 乳癌之機率=p, 不會得乳癌之機率=1- p 22 4. 男性肺腺癌患者接受化學治療一個療程後存活時間超過3 年的機率=s, <3年的機率=1-s。 Comment: In most of the cases, the probability of a concerned event differs if the definition of that event varies with different populations, time, space, or other characteristics. For example, the probability of becoming as an incidence case of breast cancer depends on the „age? of a female individual, as well as on her lifestyle, family history, genetic susceptibility, etc. Further, a male lung-cancer patient?s 3-year survival also depends on his personal characteristics; including past history of cigarette smoking or relevant exposure, occupation, genetic factors, cancer?s stage, and many others. So, the problem is complicated, which usually necessitate a complicated analysis tool and statistical „model? to explore it. Expectation and Variance (期望值與變異數) of a Bernoulli random variable: X~Ber(p) EX =1•P(X=1)+0•P(X=0) =1•p+0•(1-p)= p 22 2Var(X)=E[(X-EX)]=E(X-2•X•EX+[EX]) 2 2= E(X)- 2•EX•EX+(EX) 22=E(X)-(EX), 222but E(X)= 1•P(X=1)+0•P(X=0)= p 2?Var(X)=p-p= p(1-p) The Binomial Distribution Notation: Bin(n,p) Question If there are n independent Bernoulli trials X, =1 (probb.=p); =0 (probb.=1-p) 1 X, =1 (probb.=p); =0 (probb.=1-p) 2 …. X =1 (probb.=p); =0 (probb.=1-p) n, +) ΣX=???? i ----------------------------------------------------------- ANS: P(ΣX=0)=(1-p)(1-p)(1-p)...(1-p) i P(ΣX=1)= p (1-p)(1-p)...(1-p)+ i (1-p) p (1-p)...(1-p)+ (1-p)(1-p) p (1-p)…(1-p)+ …….+ (1-p)(1-p)...(1-p) p 1n-1 = np(1-p) 1n-1 = nC p(1-p)1 …, it can be derived similarly for other values of ΣX i Formulae xn-xP(ΣX=x) = nC p(1-p), x=0,1,2,…,n; 00 In order to check that it is a probb. distribution: -λxΣeλ/x! =1,所有的可能加起來必須等於1 x Expectation and Variance E(X)=λ and Var(X)=λ Normal (Gaussian) Distribution 常態分布,高斯分布 Figure μ μ-σ μ+σ Table μ-2σ μ+2σ Formulae 22 f(x)=[1/σ?(2π)]exp[(x-μ)/-2σ] In order to check that f(x) is a p.d.f. (probability density function), note that ? f(x)dx=1. Expectation and Variance E(X)=? xf(x)dx=μ 2Var(X)=E[(X-EX)] 2= E[(X-μ)] 2 2=E(X)-(EX) 22 = E(X)- μ 22=? xf(x)dx- μ 222 =(μ+σ) - μ 2=σ Characteristics 1. Bell-shaped,鐘形分布, 2. Symmetric about the mean μ (對稱性) 3. mean=median=mode 224. If X~Normal (μ, σ), or simply N(μ, σ), P(μ,σ,X,μ，σ) ? 0.68 P(μ,2σ,X,μ，2σ) ? 0.95 Standardization: Z=X-μ/σ 2f(z)=[1/?(2π)]exp[z/-2] (See Figures 7.8~7.11) P(-1,Z,1) ? 0.68 (exactly, =0.6826) P(-2,Z,2) ? 0.95 (exactly, =0.9544) Moreover, P(-1.96