University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution

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1 Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chi-square (χ ) distributio. t distributio. F distributio. Before we discuss the χ, t, ad F distributios here are few importat thigs about the gamma (Γ) distributio. The gamma distributio is useful i modelig skewed distributios for variables that are ot egative. A radom variable X is said to have a gamma distributio with parameters α, β if its probability desity fuctio is give by f(x) = xα 1 e x β, α, β > 0, x 0. β α Γ(α) E(X) = αβ ad σ = αβ. A brief ote o the gamma fuctio: The quatity Γ(α) is kow as the gamma fuctio ad it is equal to: Γ(α) = Useful result: 0 Γ( 1 ) = π. x α 1 e x dx. If we set α = 1 ad β = 1 λ we get f(x) = λe λx. We see that the expoetial distributio is a special case of the gamma distributio. 1

2 The gamma desity for α = 1,, 3, 4 ad β = 1. Gamma distributio desity f(x) Γ(α = 1, β = 1) Γ(α =, β = 1) Γ(α = 3, β = 1) Γ(α = 4, β = 1) x Momet geeratig fuctio of the X Γ(α, β) radom variable: Proof: M X (t) = (1 βt) α M X (t) = Ee tx = Let y = x( 1 βt ) x = β y, ad dx = β β 1 βt M X (t) = M X (t) = 1 β α Γ(α) 0 0 e tx xα 1 e x β β α Γ(α) dx = 1 x α 1 e β α Γ(α) 0 1 βt x( β ) dx dy. Substitute these i the expressio above: 1 βt ( ) α 1 β y α 1 e y β 1 βt 1 βt dy ( ) α 1 1 β β y α 1 e y dy M β α X (t) = (1 βt) α. Γ(α) 1 βt 1 βt 0

3 Theorem: Let Z N(0, 1). The, if X = Z, we say that X follows the chi-square distributio with 1 degree of freedom. We write, X χ 1. Proof: Fid the distributio of X = Z, where f(z) = 1 π e 1 z. Begi with the cdf of X: F X (x) = P (X x) = P (Z x) = P ( x Z x) F X (x) = F Z ( x) F Z ( x). Therefore: f X (x) = 1 x 1 1 π e 1 x + 1 x 1 1 e 1 x = 1 π 1 π x 1 e x, or f X (x) = x 1 e x 1 Γ( 1 ). This is the pdf of Γ( 1, ), ad it is called the chi-square distributio with 1 degree of freedom. We write, X χ 1. The momet geeratig fuctio of X χ 1 is M X (t) = (1 t) 1. Theorem: Let Z 1, Z,..., Z be idepedet radom variables with Z i N(0, 1). If Y = zi Y follows the chi-square distributio with degrees of freedom. We write Y χ. the Proof: Fid the momet geeratig fuctio of Y. Sice Z 1, Z,..., Z are idepedet, M Y (t) = M Z 1 (t) M Z (t)... M Z (t) Each Z i follows χ 1 ad therefore it has mgf equal to (1 t) 1. Coclusio: M Y (t) = (1 t). This is the mgf of Γ(, ), ad it is called the chi-square distributio with degrees of freedom. Theorem: Let X 1, X,..., X idepedet radom variables with X i N(µ, σ). It follows directly form the previous theorem that if Y = ( ) xi µ the Y χ σ. 3

4 We kow that the mea of Γ(α, β) is E(X) = αβ ad its variace var(x) = αβ. Therefore, if X χ it follows that: E(X) =, ad var(x) =. Theorem: Let X χ ad Y χ m. If X, Y are idepedet the X + Y χ +m. Proof: Use momet geeratig fuctios. Shape of the chi-square distributio: I geeral it is skewed to the right but as the degrees of freedom icrease it becomes N(, ). Here is the graph: Χ 3 f(x) x Χ 10 f(x) x Χ 30 f(x) x 4

5 Example 1 If X χ 16, fid the followig: a. P (X < 8.85). b. P (X > 34.7). c. P (3.54 < X < 8.85). d. If P (X < b) = 0.10, fid b. e. If P (X < c) = 0.950, fid c. The χ distributio - examples Example If X χ 1, fid costats a ad b such that P (a < X < b) = 0.90 ad P (X < a) = Example 3 If X χ 30, fid the followig: a. P (13.79 < X < 16.79). b. Costats a ad b such that P (a < X < b) = 0.95 ad P (X < a) = c. The mea ad variace of X. Example 4 If the momet-geeratig fuctio of X is M X (t) = (1 t) 60, fid: a. E(X). b. V ar(x). c. P (83.85 < X < ). 5

6 Theorem: Let X 1, X,..., X idepedet radom variables with X i N(µ, σ). Defie the sample variace as Proof: S = 1 1 (x i x). The ( 1)S σ χ 1. Example: Let X 1, X,..., X 16 i.i.d. radom variables from N(50, 10). Fid ( ) a. P 796. < (X i 50) < 630. ( b. P 76.1 < ) (X i X) <

7 The χ 1 (1 degree of freedom) - simulatio A radom sample of size = 100 is selected from the stadard ormal distributio N(0, 1). Here is the sample ad its histogram. [1] [5] [9] [13] [17] [1] [5] [9] [33] [37] [41] [45] [49] [53] [57] [61] [65] [69] [73] [77] [81] [85] [89] [93] [97] Histogram of the radom sample of =100 Desity z 7

8 The squared values of the sample above ad their histogram are show below. [1] e e e e+00 [5] e e e e+00 [9].9541e e e e+00 [13] e e e e-01 [17] e e e e-01 [1] e e e e-03 [5] e e e e+00 [9] 5.578e e e e-01 [33] e e e e-01 [37] e e e e+00 [41].33360e e e e-01 [45] e e e e-0 [49] e e e e-01 [53] e e e e-01 [57] e e e e-01 [61] e e e e-01 [65] e e e e-01 [69] e e e e-0 [73] e e e e-01 [77] e e e e+00 [81] e e e e-0 [85].3754e e e e-01 [89] e e e e+00 [93] e e e e-03 [97] e e e e+00 Histogram of the squared values of radom sample of =100 Desity z 8

9 The t distributio Defiitio: Let Z N(0, 1) ad U χ df. If Z, U are idepedet the the ratio Z U df follows the t (or Studet s t) distributio with degrees of freedom equal to df. We write X t df. The probability desity fuctio of the t distributio with df = degrees of freedom is f(x) = +1 Γ( ) ( ) +1 πγ( ) 1 + x, < x <. Let X t. The, E(X) = 0 ad var(x) =. The t distributio is similar to the stadard ormal distributio N(0, 1), but it has heavier tails. However as the t distributio coverges to N(0, 1) (see graph below). f(x) N(0, 1) t 15 t 5 t x 9

10 Applicatio: Let X 1, X,..., X be idepedet ad idetically distributed radom variables each oe havig N(µ, σ). We have see earlier that ( 1)S χ σ 1. We also kow that X µ σ N(0, 1). We ca apply the defiitio of the t distributio (see previous page) to get the followig: X µ σ ( 1)S σ 1 = X µ s. Therefore X µ s t 1. Compare it with X µ σ N(0, 1). Example: Let X ad SX deote the sample mea ad sample variace of a idepedet radom sample of size 10 from a ormal distributio with mea µ = 0 ad variace σ. Fid c so that P X < c = SX 10

11 Defiitio: Let U χ 1 The F distributio ad V χ. If U ad V are idepedet the ratio U 1 V follows the F distributio with umerator d.f. 1 ad deomiator d.f.. We write X F 1,. The probability desity fuctio of X F 1, f(x) = Γ( 1+ ) Γ( 1 )Γ( Mea ad variace: Let X F 1,. The, ) ( 1 is: ) 1 ( x ) 1 1 x ( 1+ ), 0 < x <. E(X) =, ad var(x) = ( 1 + ) 1 ( ) ( 4). Shape: I geeral the F distributio is skewed to the right. The distributio of F 10,3 is show below: f(x) x 11

12 Applicatio: Let X 1, X,..., X i.i.d. radom variables from N(µ X, σ X ). Let Y 1, Y,..., Y m i.i.d. radom variables from N(µ Y, σ Y ). If X ad Y are idepedet the ratio S X σ X S Y σ Y F 1,m 1. Why? Example: Two idepedet samples of size 1 = 6, = 10 are take from two ormal populatios with equal variaces. Fid b such that P ( S 1 < b) = S 1

13 Distributio related to the ormal distributio χ, t, F - summary 1. The χ distributio: Let Z N(0, 1) the Z χ 1. Let Z 1, Z,, Z i.i.d. radom variables from N(0, 1). The Zi χ. Let X 1, X,, X i.i.d. radom variables from N(µ, σ). The ( X i µ σ ) χ. The distributio of the sample variace: ( 1)S σ χ 1, where S = 1 1 (x i x), x = 1 x i Let X χ, Y χ m. If X, Y are idepedet the X + Y χ +m.. The t distributio: Let Z N(0, 1) ad U χ. Z U t. Let X 1, X,, X i.i.d. radom variables from N(µ, σ). The x µ s t 1, where S = 1 (x i x) 1, x = 1 3. The F distributio: Let U χ ad V χ m. The U V m F,m with umerator d.f., m deomiator d.f. Let X 1, X,, X i.i.d. radom variables from N(µ X, σ X ) ad Let Y 1, Y,, Y m i.i.d. radom variables from N(µ Y, σ Y ) the: x i Useful: S X σ X S Y σ Y F 1,m 1 S X = 1 1 S Y = 1 m 1 t = F 1, where (x i x), x = 1 x i m (y i ȳ), ȳ = 1 m y i ad F α;,m = 1 F 1 α;m, 13

14 Practice questios Let Z 1, Z,, Z 16 be a radom sample of size 16 from the stadard ormal distributio N(0, 1). Let X 1, X,, X 64 be a radom sample of size 64 from the ormal distributio N(µ, 1). The two samples are idepedet. a. Fid P (Z 1 > ). b. Fid P ( 16 Z i > ). c. Fid P ( 16 Z i > 6.91). d. Let S be the sample variace of the first sample. Fid c such that P (S > c) = e. What is the distributio of Y, where Y = 16 Z i + 64 (X i µ)? f. Fid EY. g. Fid V ar(y ). h. Approximate P (Y > 105). i. Fid c such that 16 c Z i F 16,80. Y j. Let Q χ 60. Fid c such that ( ) Z1 P < c = Q k. Use the t table to fid the 80 th percetile of the F 1,30 distributio. l. Fid c such that P (F 60,0 > c) =

15 Cetral limit theorem, χ, t, F distributios - examples Example 1 Suppose X 1,, X is a radom sample from a ormal populatio with mea µ 1 ad stadard deviatio σ = 1. Aother radom sample Y 1,, Y m is selected from a ormal populatio with mea µ ad stadard deviatio σ = 1. The two samples are idepedet. a. What is the distributio of W, where W is (X i X) m + (Y i Ȳ ) b. What is the mea of W? c. What is the variace of W? Example Determie which colums i the F tables are squares of which colums i the t table. Clearly explai your aswer. Example 3 The sample X 1, X,, X 18 comes from a populatio which is ormal N(µ 1, σ 7). The sample Y 1, Y,, Y 3 comes from a populatio which is also ormal N(µ, σ 3). The two samples are idepedet. For these samples we compute the sample variaces S X = (X i X) ad S Y = 1 3 (Y i Ȳ ). For what value of c does the expressio c S X S Y have the F distributio with (17, ) degrees of freedom? Example 4 Supply resposes true or false with a explaatio to each of the followig: a. The stadard deviatio of the sample mea X icreases as the sample icreases. b. The Cetral Limit Theorem allows us to claim, i certai cases, that the distributio of the sample mea X is ormally distributed. c. The stadard deviatio of the sample mea X is usually approximately equal to the ukow populatio σ. d. The stadard deviatio of the total of a sample of observatios exceeds the stadard deviatio of the sample mea. e. If X N(8, σ) the P ( X > 4) is less tha P (X > 4). 15

16 Example 5 A selective college would like to have a eterig class of 100 studets. Because ot all studets who are offered admissio accept, the college admits 1500 studets. Past experiece shows that 70% of the studets admitted will accept. Assumig that studets make their decisios idepedetly, the umber who accept X, follows the biomial distributio with = 1500 ad p = a. Write a expressio for the exact probability that at least 1000 studets accept. b. Approximate the above probability usig the ormal distributio. Example 6 A isurace compay wats to audit health isurace claims i its very large database of trasactios. I a quick attempt to assess the level of overstatemet of this database, the isurace compay selects at radom 400 items from the database (each item represets a dollar amout). Suppose that the populatio mea overstatemet of the etire database is $8, with populatio stadard deviatio $0. a. Fid the probability that the sample mea of the 400 would be less tha $6.50. b. The populatio from where the sample of 400 was selected does ot follow the ormal distributio. Why? c. Why ca we use the ormal distributio i obtaiig a aswer to part (a)? d. For what value of ω ca we say that P (µ ω < X < µ + ω) is equal to 80%? e. Let T be the total overstatemet for the 400 radomly selected items. Fid the umber b so that P (T > b) = Example 7 Next to the cash register of the Southlad market is a small bowl cotaiig peies. Customers are ivited to take peies from this bowl to make their purchases easier. For example if a customer has a bill of $.1 might take two peies from the bowl. It frequetly happes that customers put ito the bowl peies that they receive i chage. Thus the umber of peies i the bowl rises ad falls. Suppose that the bowl starts with $.00 i peies. Assume that the et daily chages is a radom variable with mea $0.06 ad stadard deviatio $0.15. Fid the probability that, after 30 days, the value of the peies i the bowl will be below $1.00. Example 8 A telephoe compay has determied that durig oholidays the umber of phoe calls that pass through the mai brach office each hour follows the ormal distributio with mea µ = ad stadard deviatio σ = Suppose that a radom sample of 60 oholiday hours is selected ad the sample mea x of the icomig phoe calls is computed. a. Describe the distributio of X. b. Fid the probability that the sample mea X of the icomig phoe calls for these 60 hours is larger tha c. Is it more likely that the sample average X will be greater tha hours, or that oe hour s icomig calls will be? 16

17 Example 9 Assume that the daily S&P retur follows the ormal distributio with mea µ = ad stadard deviatio σ = a. Fid the 75 th percetile of this distributio. b. What is the probability that i of the followig 5 days, the daily S&P retur will be larger tha 0.01? c. Cosider the sample average S&P of a radom sample of 0 days. i. What is the distributio of the sample mea? ii. What is the probability that the sample mea will be larger tha 0.005? iii. Is it more likely that the sample average S&P will be greater tha 0.007, or that oe day s S&P retur will be? Example 10 Fid the mea ad variace of S = 1 1 (X i X), where X 1, X,, X is a radom sample from N(µ, σ). Example 11 Let X 1, X, X 3, X 4, X 5 be a radom sample of size = 5 from N(0, σ). a. Fid the costat c so that c(x 1 X ) X3 + X 4 + X 5 has a t distributio. b. How may degrees of freedom are associated with this t distributio? Example 1 Let X, Ȳ, ad W ad SX, S Y, ad S W deote the sample meas ad sample variaces of three idepedet radom samples, each of size 10, from a ormal distributio with mea µ ad variace σ. Fid c so that P X + Ȳ W < c = SX + 9S Y + 9S W Example 13 If X has a expoetial distributio with a mea of λ, show that Y = X λ degrees of freedom. has χ distributio with 17

18 Example 14 Suppose that X 1, X,, X 40 deotes a radom sample of measuremets o the proportio of impurities i iro ore samples. Let each X i have a probability desity fuctio give by f(x) = 3x, 0 x 1. The ore is to be rejected by the potetial buyer if X exceeds 0.7. Fid P ( X > 0.7) for the sample of size 40. Example 15 If Y has a χ distributio with degrees of freedom, the Y could be represeted by Y = X i where X i s are idepedet, each havig a χ distributio with 1 degree of freedom. a. Show that Z = Y has a asymptotic stadard ormal distributio. b. A machie i a heavy-equipmet factory produces steel rods of legth Y, where Y is a ormal radom variable with µ = 6 iches ad σ = 0.. The cost C of repairig a rod that is ot exactly 6 iches i legth is proportioal to the square of the error ad is give, i dollars, by C = 4(Y µ). If 50 rods with idepedet legths are produced i a give day, approximate that the total cost for repairs for that day exceeds $48. Example 16 Suppose that five radom variables X 1,, X 5 are i.i.d., ad each has a stadard ormal distributio. Determie a costat c such that the radom variable c(x 1 + X ) X3 + X 4 + X 5 will have a t distributio. Example 17 Suppose that a radom variable X has a F distributio with 3 ad 8 degrees of freedom. Determie the value of c such that P (X < c) = Example 18 Suppose that a radom variable X has a F distributio with 1 ad 8 degrees of freedom. Use the table of the t distributio to determie the value of c such that P (X > c) = 0.. Example 19 Suppose that a poit (X, Y, Z) is to be chose at radom i 3-dimesioal space, where X, Y, ad Z are idepedet radom variables ad each has a stadard ormal distributio. What is the probability that the distace from the origi to the poit will be less tha 1 uit? Example 0 Suppose that X 1,, X 6 form a radom sample from a stadard ormal distributio ad let Y = (X 1 + X + X 3 ) + (X 4 + X 5 + X 6 ). Determie a value of c such that the radom variable cy will have a χ distributio. 18