Let X and Y be independent positive random variables. Let Z=X/Y. In what follows, all occurrences of x, y, z are assumed to be positive numbers.
1) Suppose that X and Y are discrete, with known PMFs, p_X and p_Y. Then, pZ|Y(z|y)=pX(?).
What is the argument in the place of the question mark?
2) Suppose that X and Y are continuous, with known PDFs, f_X and f_Y. Provide a formula, analogous to the one in part (a), for f_Z|Y(z|y) in terms of f_X. That is, find A and B in the formula below.
f_Z|Y(z|y)=Af_X(B).
A=
B=
3)Which of the following is a formula for f_Z(z)?
f_Z(z)=…
(Choose all that apply.)
to see the options :
I[dot]imgur[dot]com/WQOtsyh.jpg
1. z*y
2. A = ?
B = ?
3. Option 4
To find the answers to the given questions, we need to apply the concepts of conditional probability and probability density functions (PDFs). Let's go through each question step by step:
1) For discrete random variables X and Y, the formula for conditional probability is given by:
pZ|Y(z|y) = p(X = zy | Y = y)
In this case, we are looking for the argument in the place of the question mark, which represents the value of X that satisfies the condition Z = X/Y = z given Y = y.
To find pX(?), we need to consider the joint probability mass function (PMF) of X and Y, which gives the probability of X = x and Y = y happening simultaneously. We can express this as:
pX(? | Y = y) = p(X = ? | Y = y)
So, the argument in the place of the question mark is simply "?".
2) For continuous random variables X and Y, the formula for conditional probability density function (PDF) is given by:
fZ|Y(z|y) = (1 / |B|) * fX(Bz)
In this case, we are looking for the values of A and B. A represents the normalization constant, and B determines how Z and Y are related.
A = 1 / |B|
B = Y
So, the formula becomes:
fZ|Y(z|y) = (1 / Y) * fX(zY)
3) To determine the formula for fZ(z), we consider the case when X and Y are independent, meaning there is no conditional relationship. In this case, we can simply multiply the PDFs of X and Y to get the PDF of Z.
The options provided in the image are:
(a) fX(z) * fY(z)
(c) fX(z / y) * fY(y)
Both options (a) and (c) are correct formulas for fZ(z)
1) The argument in the place of the question mark in the given expression pZ|Y(z|y)=pX(?) is y/z.
2) The formula for f_Z|Y(z|y) in terms of f_X is:
f_Z|Y(z|y) = (1/y) * f_X(z/y)
where A = 1/y and B = z/y.
3) The formula for f_Z(z) can be given as:
f_Z(z) = ∫[0,∞] f_X(zy) * f_Y(y) * |1/y| dy
The options mentioned in the image are not visible, so I cannot provide a specific answer.