PART A: complete the two-way frequency table

For each given p, let ???? have a binomial distribution with parameters p and ????. Suppose that ???? is itself binomially distr

pshichka [43]

Answer:

See the proof below.

Step-by-step explanation:

Assuming this complete question: "For each given p, let Z have a binomial distribution with parameters p and N. Suppose that N is itself binomially distributed with parameters q and M. Formulate Z as a random sum and show that Z has a binomial distribution with parameters pq and M."

Solution to the problem

For this case we can assume that we have N independent variables $X_i$ with the following distribution:

$X_i Bin (1,p) = Be(p)$ bernoulli on this case with probability of success p, and all the N variables are independent distributed. We can define the random variable Z like this:

$Z = \sum_{i=1}^N X_i$

From the info given we know that $N \sim Bin (M,q)$

We need to proof that $Z \sim Bin (M, pq)$ by the definition of binomial random variable then we need to show that:

$E(Z) = Mpq$

$Var (Z) = Mpq(1-pq)$

The deduction is based on the definition of independent random variables, we can do this:

$E(Z) = E(N) E(X) = Mq (p)= Mpq$

And for the variance of Z we can do this:

$Var(Z)_ = E(N) Var(X) + Var (N) [E(X)]^2$

$Var(Z) =Mpq [p(1-p)] + Mq(1-q) p^2$

And if we take common factor $Mpq$ we got:

$Var(Z) =Mpq [(1-p) + (1-q)p]= Mpq[1-p +p-pq]= Mpq[1-pq]$

And as we can see then we can conclude that $Z \sim Bin (M, pq)$

8 0

3 years ago

Record the product 94 x 12 =

Ahat [919]

Answer:

1128

Step-by-step explanation:

Math

3 0

2 years ago

OBTAINING ONE HEAD ONLY WHEN A DIE IS<br> ROLLED AND 2 COINS ARE TOSSED.<br> Probability

Valentin [98]

Answer:

⅓

Step-by-step explanation:

n(s) 1 die & 2 coins = 6×4 = 24

n (event) 1 head = 6 (by die) +2( by coins)

= 8

so, probability P = 8/24 = ⅓

8 0

3 years ago

Which statement describes the graph of y + 2 ≥ –4(x – 3)2? a parabola opening up, with shading above the vertex a parabola openi

olasank [31]

Option C. The statement that describes the graph of the equation y + 2 ≥ –4(x – 3)2 is a a parabola opening down, with shading above the vertex.

<h3>How to describe the shape of the parabola.</h3>

To do this you have to create a graph of the equation as seen in the attachment.

From the attachment we can see that the parabola is shaped downwards while we have the shading to be at the top of the vertex.