## practice problems for final exam

概率导论final exam代写 How many ways are there to divide 5 different cakes and 5 identical cookies between 2 people so that the each person gets exactly 5 items.

*1.(a) How many ways **are **there **to divide 5 different **cakes **and 5 identical **cookies between **2 **people** **so** **that** **the** **first** **person** **gets** **exactly** **3 **cakes. 概率导论final exam代写*

*(b) How many ways are there to divide 5 different cakes and 5 identical cookies between 2 **people so that the each person gets exactly 5 items.*

*2.There **are **three ** The first contains 2 **red **balls and 5 blue balls, the **second **contains 3 **red **balls and 4 blue balls and the **third **contains 4 **red **balls and 3 blue balls. **An **urn is**chosen at random and then balls **are **drawn **without** **replacement.*

*(a)Find the conditional **probability **that the **third **urn was chosen given that the first **three **draws resulted **in 2 **red **and 1 blue **ball.*

*(b)Find**the** **conditional** **probability** **that** **the** **fourth** **ball** **will** **be** **blue** **given** **that** **the** **first** **three **draws resulted **in 2 **red **and 1 blue **ball.*

*3.(a) 3 **people **get 3 **cards **each from **the standard 52 **card ** **Find the **probability**that at** **least **one person has only **red** **cards.*

*(b) 20 tables have 3 palyers and one standard 52 **card **deck **per **table. **At each **table **each **player is given 3 **cards **from **the **deck. **Let **X **be **the number of tables **where **at **least **one player **has only **red **cards. **Compute **P *(*X *= 6)*.*

*4.6 **dice **are * *Let **X **be **the number of different numbers which **occur **(for example if the numbers shows **are **6, 2, 3, 6, 6, 3 then **X *=3*).*

*(a)Compute **P *(*X *=3)*.*

*(b)Compute EX.*

*5.Let **X **have denisty **xe*^{−}^{x}*if **x *0 *and 0 **otherwise. 概率导论final exam代写*

^{−}

^{x}

*(a)Compute **P *(*X >*2)*.*

*(b)Find the density of **Y *=*X*^{2}*.*

*6.When a crush **occurs **on 40 mile highway its **location **has density*^{x}* **.*

*(a)Find a **probability **that the next crust **will **occur **between **mileposts 10 and 30. *

*(b)During**a** **certain** **week** **there** **was** **10** ** **Find** **the** **probability** **that** **exactly** **6** **occurred **between mileposts 10 and 30.*

*(c)Let **N **be **the first crush which **appear before **milepost 10. Compute **EN **and **VN.*

*(d)Where **on highway should a service station **be located **to minimize the **expected**distance to the next** **crush?*

*7.Let *(*X, Y *) *have density **equal **to **cxy**if** **x* 0*,** **y* 0 *and **x** *+ *y* 1 *and **equal **to 0 otherwise. 概率导论final exam代写*

*(a)Compute **E*(*XY*)*.*

*(b)Compute **P *(*X > *2*Y*)*.*

*(c)Compute the marginal distribution of **X.*

*8.Let **X **and **Y **be **independent, **X **have uniform distribution on *(0*, *1) *and **Y **have uniform distribution on *(0*,*2)*.*

*(a)Find the density of **Z *= *X *+Y.

*(b)Find **P *(*X > Y*)*.*

*9.Let *(*X, Y *) *be **a random variables with **EX *= *EY *= 1*, V *(*X*) = 5*, V *(*Y *) =10*,**Cov*(*X, Y *) = 2*.*

*(a)Compute Cov*(*X *+*Y,** **X* *Y** *)*.*

*(b)0Compute **V *(2*X *+ 3*Y*)*.*

*(c)Compute **E*(2*X *+ 3*Y*)^{2}*.*

*(d)Use**one** **sided** **Chebyshev** **inequality** **to** **estimate** **P** *(2*X** *+ 3*Y** **>** *30)*.*

*10.Let **X **and **Y **denote tax revenue **from **individuals and busynesses * *Suppose that*(*X,** **Y** *) *is** **a** **normal** **vector** **and** **EX** *= 15*,** **EY** *= 20*,** **V** *(*X*) = 1*,** **V** **Y** *= 9*,** **Cov*(*X,** **Y** *)*=2.*

*(a)Let **T *= *X *+ *Y **. Find the distribution of **T.*

*(b)Let **D *= *Y **− **X**. **Find **P *(*D <*4)*.*

*(c)Compute **P *(*X < *15*|**Y *=21)*.*

*(d)Compute **P *(*X < *15*|**T *=33)*.*

*11.Let **N **be **the number of times Jane stops at traffic lights during her drive to and **from **work** **Suppose** **that** **P** *(*N** *= 1) = 0*.*2*,** **P** *(*N** *= 2) = 0*.*5*,** **P** *(*N** *= 3) = 0*.*3*.** **Suppose **moreover **that the time of **each **stop is uniformly distributed on the interval **from **0 to 2 minutes and that those times **are **independent. **Let **T **be **the total number Jane waits at the traffic light during her work** **day. 概率导论final exam代写*

*(a)Compute **E*(*T **N *= 2) *and **V *(*T N *=2)*.*

*(b)Compute **E*(*T *) *and **V *(*T*)*.*

*(c)Let **M **be **the total number of times Jane is **stopped **at the traffic light on Monday and * *Compute **P *(*M *=4)*.*

*(d)Let **Z **be **the total time Janes waits at traffic lights during a particular workweek (Monday-Friday). Compute **EZ.*

*(e)Let **D **be **the total number of days during a workweek Jane has exactly **three ** Compute **P *(*D *=2)*.*

*(f) **Call **a **week **unlucky if Jane has **three **traffic light stop **per **day for at **least **three **days.*

*Let **L **be the number of the first unlucky week. Compute **EL.*

*12.Let **X*_{1}*, **X*_{2}*. . . X*_{240} *be **independent identically distributed random variables such that** **X*_{j }*has density *3*x*^{2} *if **x *[0*, *1] *and 0 otherwise. **Let **S *= *X*_{1} + *X*_{2} + + *X*_{240}*. 概率导论final exam代写*

_{j }

*(a)Compute **ES **and **VS.*

*(b)Find approximately **P *(*S >*183)*.*

*13.Misprints in a 400 **page **book **form a Poisson **process **with intensity 3**misprints/page.*

*(a)Let **X **be **the total number of misprints in a * *Compute **EX **and **V**X.*

*(b)Compute approximately**P** *(*X* 1150)*.*

*Let **N **be **the first **page **without misprints. Compute **P *(*N *=25)*.*

*(c)Compute a **probability **that in a 30 **page **chapter **there **are **exactly 2 misprint **free **pages.*

*The **book **is **read **by two * *Each misprint is **captured **by **each **proofreader **with **probability *0*.*8 *independently of the other **proofreader.** **Let **Y **be **the total number** **of misprint** **left** **in** **the** **book** **after** **the** **proofreading.** **Compute** **EY** **and** **V** **Y.*

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