Mathematics, 28.07.2019 21:30 manarsadi6
Let y have a poisson distribution with mean λ. find e [y (y − 1)] and then use this to show that v(y) = λ.
Answers: 1
follows a Poisson distribution with mean 
, we know the PMF of 
![\mathbb E[Y(Y-1)]=\mathbb E[Y^2-Y]=\mathbb E[Y^2]-\mathbb E[Y]](/tpl/images/0144/2713/3e15b.png)
![\implies\mathbb E[Y(Y-1)]=\mathbb E[Y^2]-\lambda](/tpl/images/0144/2713/1654b.png)
![(\mathbb E[Y^2])](/tpl/images/0144/2713/60209.png)
, we can use the definition of the expectation of a function of a random variable:![\mathbb E[Y^2]=\displaystyle\sum_yy^2\,f_Y(y)=\sum_{y=0}^\infty y^2\frac{e^{-\lambda}\lambda^y}{y!}](/tpl/images/0144/2713/7facb.png)
![\mathbb E[Y^2]=\displaystyle\sum_{y=1}^\infty y^2\frac{e^{-\lambda}\lambda^y}{y!}](/tpl/images/0144/2713/be5c2.png)
![\mathbb E[Y^2]=\displaystyle\lambda\sum_{y=1}^\infty y\frac{e^{-\lambda}\lambda^{y-1}}{(y-1)!}](/tpl/images/0144/2713/02766.png)
![\mathbb E[Y^2]=\displaystyle\lambda\sum_{y=0}^\infty(y+1)\frac{e^{-\lambda}\lambda^y}{y!}](/tpl/images/0144/2713/fef8b.png)
![\mathbb E[Y^2]=\displaystyle\lambda\left(\sum_{y=0}^\infty y\frac{e^{-\lambda}\lambda^y}{y!}+\sum_{y=0}^\infty\frac{e^{-\lambda}\lambda^y}{y!}\right)](/tpl/images/0144/2713/d01e5.png)
![\implies\mathbb E[Y^2]=\lambda(\mathbb E[Y]+1)=\lambda^2+\lambda](/tpl/images/0144/2713/3b024.png)
![\mathbb E[Y(Y-1)]=\lambda^2+\lambda-\lambda=\lambda^2](/tpl/images/0144/2713/85c24.png)
![\mathbb V[Y]=\mathbb E[(Y-\mathbb E[Y])^2]=\mathbb E[Y^2]-\mathbb E[Y]^2](/tpl/images/0144/2713/02344.png)
![\mathbb V[Y]=\underbrace{\mathbb E[Y^2]-\mathbb E[Y]}_{\mathbb E[Y(Y-1)]}+\mathbb E[Y]-\mathbb E[Y]^2](/tpl/images/0144/2713/4b40e.png)
![\implies\mathbb V[Y]=\lambda^2+\lambda-\lambda^2](/tpl/images/0144/2713/71af0.png)
![\implies\mathbb V[Y]=\lambda](/tpl/images/0144/2713/2ac7f.png)
Another question on Mathematics
Mathematics, 21.06.2019 23:00
What is the location of point g, which partitions the directed line segment from d to f into a 5: 4 ratio? –1 0 2 3
Answers: 1
Mathematics, 21.06.2019 23:40
Statement reason 1. δabc is similar to δced. given 2. 3. definition of slope 4. slope of slope of definition of slope 5. slope of × slope of multiplying the slopes 6. slope of × slope of substitution property of equality 7. slope of × slope of simplifying the right side the table contains the proof of the relationship between the slopes of two perpendicular lines. what is the reason for statement 2? a. parallel line segments that meet a common perpendicular line are proportional in length. b. the lengths of vertical and horizontal sides in congruent triangles are in a common ratio. c. trigonometric identities determine the lengths of the legs in a right triangle. d. corresponding side lengths in similar triangles are proportional in length.
Answers: 2
Mathematics, 22.06.2019 01:30
This graph shows a portion of an odd function. use the graph to complete the table of values. x f(x) −2 −3 −4 −6
Answers: 3
Mathematics, 22.06.2019 02:30
Anne plans to increase the prices of all the items in her store by 5%. to the nearest cent, how much will an artist save if the artist buys a canvas and a frame that each measure 24 by 36 inches before the price increase goes into effect?
Answers: 1
You know the right answer?
Let y have a poisson distribution with mean λ. find e [y (y − 1)] and then use this to show that v(y...
Questions
Mathematics, 19.04.2021 20:50
Mathematics, 19.04.2021 20:50
History, 19.04.2021 20:50
English, 19.04.2021 20:50
Mathematics, 19.04.2021 20:50
History, 19.04.2021 20:50
Chemistry, 19.04.2021 20:50
French, 19.04.2021 20:50
History, 19.04.2021 20:50
