Let ? be a positive integer random variable with PMF of the form ??(?)=12⋅?⋅2−?,?=1,2,…. Once we...

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Question

Let ? be a positive integer random variable with PMF of the form ??(?)=12⋅?⋅2−?,?=1,2,…. Once we...

Let ? be a positive integer random variable with PMF of the form

??(?)=12⋅?⋅2−?,?=1,2,….

Once we see the numerical value of ?, we then draw a random variable ? whose (conditional) PMF is uniform on the set {1,2,…,2?}.

1.1

Write down an expression for the joint PMF ??,?(?,?).

For ?=1,2,… and ?=1,2,…,2?:

??,?(?,?)=?

1.2

Find the marginal PMF ??(?) as a function of ?. For simplicity, provide the answer only for the case when ? is an even number. (The formula for when ? is odd would be slightly different, and you do not need to provide it).

Hint: You may find the following helpful: ∑?=0∞??=11−? for 0<?<1.

For ?=2,4,6,…:

??(?)=?

1.3.

Let ? be the event that ? is even. Find ?(?|?=?) and ?(?).

?(?∣?=?)= ?

?(?)=?

1.4 Is the event ? independent of ??

math Statistics-And-Probability

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Answer 1

Answer #1

Sell Pcnonga tops, =1,2,-- Orow [Pinon) - 1 • $[$tent en to-.Jul Az{k is even} K-2n, ie K= 2,4,6,- Plkshya , & ,1,2,--. 0 T

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Answer 2

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Let ? be a positive integer random variable with PMF of the form ??(?)=12⋅?⋅2−?,?=1,2,…. Once we...

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Let ? be a positive integer random variable with PMF of the form ??(?)=12⋅?⋅2−?,?=1,2,…. Once we...

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Let ? be a positive integer random variable with PMF of the form ??(?)=12⋅?⋅2−?,?=1,2,…. Once we...

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Add Answer to: Let ? be a positive integer random variable with PMF of the form ??(?)=12⋅?⋅2−?,?=1,2,…. Once we...

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Let ? be a positive integer random variable with PMF of the form ??(?)=12⋅?⋅2−?,?=1,2,…. Once we...