Consider two discrete random variables x, y , taking values in {0, 1, 2, 3} each (for a total of 16 possible points). their joint probability mass function is given by fxy (x, y) = c · x+1 y+1 . answer the following questions. (a) what is c? (b) what is the probability that x = y ? (c) derive the marginal probability mass functions for both x and y . (d) what is the probability that x ≤ y given that y = 2? (e) what is the probability that x ≤ y ?

A. c=1/100

B. P(x==y)=3/10

C.

D. P(x≤y | y=2)=9/50

E. P(x≤y)= 7/20

Step-by-step explanation:

A. To answer the question we use a really important property about mass functions of probability: If is a probability mass function, then (for a discrete variables because that´s our case)

So, for our problem we have:

And we solve for c:

Because x and y only takes values in [0,1,2,3] we know:

And we use that to solve for c:

Finally we have c=1/100

B. For part B, we just choose the probabilities that we want because there are few values to choose and use the mass function to find them:

P(x=0,y=0)=(0+1)(0+1)/100=1/100

P(x=1,y=1)=4/100 (putting x=1 and y=1 in the function)

P(x=2,y=2)=9/100

P(x=3,y=3)=16/100

And for the last part we add them:

P(X==Y)=1/100+4/100+9/100+16/100=3/10

This is the answer

C. We find (in discrete variables) the marginal probability mass function just solving the function for one of the variables as follows:

And we do the same for y:

These functions are the marginal probability mass functions for x and y

D. Because there are few values of x and y that they can be. we choose the probabilities as follows:

P(x=0,y=2)=(0+1)(1+2)/100=3/100

P(x=1,y=2)=6/100

P(x=2,y=2)=9/100

And we add them to find our answer

P(x≤y | y=2)=3/100+6/100+9/100=9/50

E. We are working with a few amount of possible values, so we can use the part B (P(x==y)) and add them the probabilities that are left to complete P(x≤y)

P(x==y)=3/10 (B)

P(x=0,y=1)=2/100

P(x=0,y=2)=3/100

P(x=1,y=2)=6/100

P(x=0,y=3)=4/100

P(x=1,y=3)=8/100

P(x=2,y=3)=12/100

We add them to get our final

P(x≤y)=7/20

are ya tryin to ask something or is this a statement

an = 22 + (n-1) *6

110 seats

step-by-step explanation:

an=a1+(n-1)(d)

a1 = 22   this is the first item in the sequence

d =6   this is how the sequence goes up

a)   an = 22 + (n-1) *6

b) a12 = 22 + (12-1) *6

    a12 = 22+11*6

  a12 = 22 +88

  a12 = 110