Uniform Distribution. Uniform Distribution. Uniform Distribution. Graphs of Gamma Distributions. Gamma Distribution. Continuous Distributions CD - 1

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1 Continuou Uniform Special Probability Denitie Definition 6.: The continuou random variable ha a uniform ditribution if it p.d.f. i equal to a contant on it upport. If the upport i the interval [a b] then it p.d.f. i The mot common parameter in mot of the ditribution are the lower moment: mean m and variance. f ( x) a x b. b a It i uually denoted a U(a b). f(x) F(x) b a 0 a b 0 a b Uniform Uniform The mean variance and m.g.f. of a continuou random variable that ha a uniform ditribution are: : Let be U(00) find the mean and the variance and the m.g.f. of. a b m tb ta e e M ( t ) t ( b a ) t 0 t 0. ( b a ) Peudo-Random Number Generator on mot computer U(0 ) m 5 0 t e M ( t ) 0 t (0 0) t 0 t Gamma Graph of Gamma Definition 6.: The continuou random variable ha a Gamma ditribution Gamma( a b ). if it p.d.f. i f ( x) x a ( a ) b 0 a e x / b for elewhere. Gamma Function: (n) = (n )! 0 x * can be the waiting time until ath ucce in a Poion proce. 5 a = /4 a = a = a = 3 a = b = 4 a = 4 b = 5/6 b = b = b = 3 b = CD -

2 Continuou Gamma The mean variance and m.g.f. of a continuou random variable that ha an Gamma ditribution are: m ab M ( t ) ( b t ) a ab t b Gamma( b ) Exponential Gamma( r/ ) Ch-quare with d.f. = r. 7 8 Special Notation c a (r) Let be a random that ha Chi-quare ditribution with degree of freedom r P[ c a ( r )] a Tail area Probability of greater than or equal to c a(r) i a. Find c.0(7) =? Try Thi! Find c.05(4) =? Find the 0 th percentile from a c ditribution with degree of freedom 6. P[ r c a ( )] a c a(r) : Find c.05(3) =? Exponential Definition 6.3 The continuou random variable ha an exponential ditribution if it p.d.f. i Exponential The mean variance and m.g.f. of a continuou random variable that ha an exponential ditribution are: x x f ( x) 0 / for 0 x elewhere where i the mean of the ditribution. m M ( t ) t t * can be the waiting time until next ucce in a Poion proce. c.d.f. 0 F x) x e ( / x 0 0 x CD -

3 Continuou Exponential : Let have an exponential ditribution with a mean of 30 what i the firt quartile of thi ditribution? F ( p.5 0 F x) x / e ( 30 ) e p / x 0 0 x p e.75 p.5 / = ln(.75) p.5 = ln(.75) { = 30} p.5 = - 30 ln(.75) p.5 = 8.63 / Exponential Let W be the waiting time until next ucce in a Poion proce in which the average number of ucce in unit interval i l then for w 0 F(w) = P(W w) = P(W > w) = P(W > w) = P(no ucce in [0w]) = e -lw d.f. of exponential ditribution. F (w) = f(w) = le -lw p.d.f. of exponential ditribution. x / p 3 p = ln( p) e 4 Exponential Let W be the waiting time until next ucce in a Poion proce in which the average number of ucce in unit interval i l then l = / where i the average waiting time until next ucce for w 0 Exponential Suppoe that number of arrival of cutomer follow a Poion proce with a mean of 0 per hour. What i the probability that the next cutomer will arrive within 5 minute? (5 min. = 0.5 hour) F(w) = e -lw = e -w/ f(w) = e w/ d.f. of exponential ditribution. p.d.f. of exponential ditribution. 5 P( x) = F (x) = e -x/ P( 0.5) = F (0.5) = e -0.5/ l = 0 = 0. F (0.5) = e -0.5/0. =.98 6 Beta Definition 6.5: The continuou random variable ha a Beta ditribution Beta( a b ). if it p.d.f. i Beta The mean and variance of a continuou random variable that ha an Beta ditribution are: f ( x) ( a b ) x ( a ) ( b ) 0 a ( x) b for 0 x elewhere. a m a b ab ( a b ) ( a b ) for a > 0 and b > 0. Beta function : Γ(α)Γ(β) = Γ(α+β) xα ( x) β dx CD - 3

4 Continuou Probability Denity Function Definition 6.6: The continuou random variable ha a normal ditribution if it p.d.f. i m The mean variance and m.g.f. of a continuou random variable that ha a normal ditribution are: f ( x) e p x m < x < E[ ] m M ( t ) e m t t / Var [ ] Notation: N (m ) A normal ditribution with mean m and tandard deviation 9 0 m.g.f. of : If the m.g.f. of a random variable i M(t) = exp(t + 6t ) what i the ditribution of thi random variable? What are the mean and tandard deviation of thi ditribution and what i the p.d.f. of thi random variable?? Mean? Standard deviation? p.d.f.?. Bell-Shaped & Symmetrical. Mean Median Mode Are Equal 3. Random Variable Ha Infinite Range < x < f() Mean Median Mode Effect of Varying Parameter (m & ) N (7 5) A normal ditribution with mean 7 and variance 5. Poible ituation: Tet core pule rate f(x) B N (30 4) A normal ditribution with mean 30 and variance 4. Poible ituation: Weight Choleterol level A C x 3 4 CD - 4

5 Continuou f() Infinite Number of Table ditribution differ by mean & tandard deviation. Each ditribution would require it own table. Standard Standard : Definition 6.7: A normal ditribution with mean = 0 and tandard deviation = i referred to a the Standard. = Notation: That an infinite number! 5 ~ N (m = 0 = ) Cap letter 0 6 Area under Standard Curve 0 z How to find the proportion of the are under the tandard normal curve below z or ay P ( < z ) =? Ue Standard Table!!! 7 8 Standard P( < 0.3) = F(0.3) Area below.3 = 0.655? Standard P( > 0.3) = Area above.3 =.655 =.3745 Area in the upper tail of the tandard normal ditribution CD - 5

6 Continuou Standard P(0 < < 0.3) = Area between 0 and.3 =? = = 0.55 P ( -.00 < <.00 ) =? P ( -.00 < <.00 ) =.9544 P ( 3.00 < < 3.00 ) = = P (.40 < <.33 ) =.9093 = Standard m = = 3 = m = 50 m = CD - 6

7 Continuou Standardize the m m Standardized m= 0 One table! = 37 Theorem 6.7: If ha a normal ditribution with mean m and variance μ then = σ ha a tandard normal ditribution. P x < < x = x e x μ σ dx πσ z = x μ σ z = x μ σ = π = x z e z z z π e z z = P z < < z dx dx 38 Standardize the a N ( m ) m b a m N ( 0 ) 0 Standard b m = 0 For a normal ditribution that ha a mean = 5 and.d. = 0 what percentage of the ditribution i between 5 and 6.? a m b m P ( a b) P 39 m = = 0 m m P(5 6.) P(0.) Standardized = Standardized Table = m= 0. m= 5 6. m= 0. 4 Area = = CD - 7

8 Continuou = 0 m m P(5 6.) P(0.) % Standardized = = 0 P(3.8 5) m P(3.8 5) =P(. 0).0478 Standardized =.0478 m= 5 6. m= m = 5 -. m = 0 Area = P( > 8) P( > 8) = 0 m P( > 8) =P( >.30) =.38 Standardized =.38 = 0 6% 38% Value 8 i the 6 nd percentile m = 5 8 m = 0 Area = m = More on The work hour per week for reident in Ohio ha a normal ditribution with m = 4 hour & = 9 hour. Find the percentage of Ohio reident whoe work hour are A. between 4 & 60 hour. P(4 60) =? B. le than 0 hour. P( 0) =? = 00 9 P(4 60) =? m m Standardized P(4 60) P(0 ) =.477(47.7%) CD - 8

9 Continuou P( 0) =? Finding Value for Known Probabilitie m What i z given P( < z) =.80? Standardized Table = 00 9 P( 0) = P(.44) = = 0.73% Standardized = z =.84 z.0 =.84 Def. z a : P( z a ) = a ; P( < z a ) = a 50 Finding Value for Known Probabilitie Finding Value for Known Probabilitie : The weight of new born infant i normally ditributed with a mean 7 lb and tandard deviation of. lb. Find the 80th percentile. Area to the left of 80th percentile i In the table there i a area value (cloe to 0.800) correponding to a z-core of th percentile = x. = lb 5 Standardized = z = th percentile m 7(.84)(. ) More The pule rate for a certain population follow a normal ditribution with a mean of 70 per minute and.d. 5. What percent of thi ditribution that i in between 60 to 80 per minute? 6.6 Approximation for Binomial Probability P ( 3)? b m = np = 3.5 = np(p) =.75 The weight of a population follow a normal ditribution with a mean 30 and.d. 0. What percent of thi population that i in between 0 and 50 lb? CD - 9

10 Continuou P ( 3) P ( >.5) b P ( > ).75 P ( >.76 ).7764 Binomial Table: P( b 3) =.7734 Approximation for Binomial Probability N m = np = 3.5 = np(p) = The Bivariate Definition 6.8: The pair of random variable and Y have a bivariate normal ditribution if and only if their joint probability denity function i given by f x y = e ρ x μ σ ρ x μ σ y μ σ + y μ σ for < x < and πσ < σy < where ρ > 0 > 0 and r <. for < x < and < y < where > 0 > 0 and r <. 56 cov( Y) r r cov( Y) / ( ) i the correlation coefficient. Theorem 6.9: If and Y have a bivariate normal ditribution the conditional denity of Y given = x i a normal ditribution with Theorem 6.0: If and Y have a bivariate normal ditribution they are independent if and only if r = 0. μ Y x =μ + ρ σ σ (x μ ) σ Y x = σ ( ρ ) μ y =μ + ρ σ σ (y μ ) σ y = σ ( ρ ) CD - 0