In Poisson probability distribution if P(r=2)= 3P(r=3), and P(r=4) is given by Ae/24B24/eС1/(24e)D24e About the author Cora
[tex]\large\underline{\sf{Solution-}}[/tex] We know, Probability of any random variable ‘r’ having mean m using Poisson Distribution is given by [tex]\rm :\longmapsto\:P(r) = \dfrac{ {e}^{ – m} \: \: {m}^{r} }{r! } [/tex] According to statement, It is given that P(2) = 3 P(3) [tex]\rm :\longmapsto\:\dfrac{ \cancel{ {e}^{ – m} }\: {m}^{2} }{2!} = 3 \dfrac{ \cancel{{e}^{ – m} }\: {m}^{3} }{3!} [/tex] [tex]\rm :\longmapsto\:\dfrac{1}{2 \times 1} = 3 \times \dfrac{m}{3 \times 2 \times 1} [/tex] [tex]\bf\implies \:m \: = \: 1[/tex] Now, [tex]\rm :\longmapsto\:P(4)[/tex] [tex]\rm \: \: = \: \dfrac{ {e}^{ – m} \: {m}^{4} }{4!} [/tex] [tex]\rm \: \: = \: \dfrac{ {e}^{ – 1} {(1)}^{4} }{4!} [/tex] [tex] \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \red{\bigg \{ \because \:m \: = \: 1 \bigg \}}[/tex] [tex]\rm \: \: = \: \dfrac{ {e}^{ – 1} }{4 \times 3 \times 2 \times 1} [/tex] [tex]\rm \: \: = \: \dfrac{1}{24e} [/tex] Hence, [tex]\bf :\longmapsto\:P(4) \: = \: \dfrac{1}{24e} [/tex] Reply
[tex]\large\underline{\sf{Solution-}}[/tex]
We know,
Probability of any random variable ‘r’ having mean m using Poisson Distribution is given by
[tex]\rm :\longmapsto\:P(r) = \dfrac{ {e}^{ – m} \: \: {m}^{r} }{r! } [/tex]
According to statement,
It is given that
P(2) = 3 P(3)
[tex]\rm :\longmapsto\:\dfrac{ \cancel{ {e}^{ – m} }\: {m}^{2} }{2!} = 3 \dfrac{ \cancel{{e}^{ – m} }\: {m}^{3} }{3!} [/tex]
[tex]\rm :\longmapsto\:\dfrac{1}{2 \times 1} = 3 \times \dfrac{m}{3 \times 2 \times 1} [/tex]
[tex]\bf\implies \:m \: = \: 1[/tex]
Now,
[tex]\rm :\longmapsto\:P(4)[/tex]
[tex]\rm \: \: = \: \dfrac{ {e}^{ – m} \: {m}^{4} }{4!} [/tex]
[tex]\rm \: \: = \: \dfrac{ {e}^{ – 1} {(1)}^{4} }{4!} [/tex]
[tex] \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \red{\bigg \{ \because \:m \: = \: 1 \bigg \}}[/tex]
[tex]\rm \: \: = \: \dfrac{ {e}^{ – 1} }{4 \times 3 \times 2 \times 1} [/tex]
[tex]\rm \: \: = \: \dfrac{1}{24e} [/tex]
Hence,
[tex]\bf :\longmapsto\:P(4) \: = \: \dfrac{1}{24e} [/tex]