Course: Mathematics-I (CDA101), Topic- Function of independent Random
Variables, Covariance, Moments
Tutorial Sheet #6
1. Find the PDF of eX in terms of the PDF of X . Specialize the answer to the case
where X is uniformly distributed between 0 and 1.
2. If X is a√random variable that is uniformly distributed between −1 and 1. find the
PDF of X and the PDF of − ln |X|.
3. Find the PDF of Z = X + Y, Z= X-Y, Z=XY, and Z=X/Y in terms of PDF of X
and Y, when X and Y are independent random variables.
4. For moment generating function of random variable X, prove that:
MaX (t) = MX (at),
MX+a (t) = eat MX (t).
5. Find the covariance of random variables X and Y having the joint probability density
function
x + y : 0 < x < 1, 0 < y < 1
f (x, y) =
0 : otherwise
6. Suppose that X and Y are independent random variables with probability densities
8/x3 :x>2
g(x) =
0 : otherwise
2y :0<y<1
h(y) =
0 : otherwise
Find the expected value of Z = XY.
7. Let X represent the number that occurs when a red die is tossed and Y the number
that occurs when a green die is tossed. Find E[(X+Y)], E[(X-Y)], E[(XY)].
8. A random variable X has the discrete uniform distribution
1/k : x = 1, 2, ..., k
f (x; k) =
0 : otherwise
Find moment generating function of X.
9. A random variable X has the geometric distribution g(x; p) = pq x−1 for x = 1, 2,
3,... Find moment-generating function of X and then use MX (t) to find the mean
and variance of the geometric distribution.
−λ x
10. A random variable X has the Poisson distribution p(x; λ) = e x!λ for x = 0, 1, 2,...
Find moment-generating function of X and then use MX (t) to find the mean and
variance of the Poisson distribution.
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Variables, Covariance, Moments
Tutorial Sheet #6
1. Find the PDF of eX in terms of the PDF of X . Specialize the answer to the case
where X is uniformly distributed between 0 and 1.
2. If X is a√random variable that is uniformly distributed between −1 and 1. find the
PDF of X and the PDF of − ln |X|.
3. Find the PDF of Z = X + Y, Z= X-Y, Z=XY, and Z=X/Y in terms of PDF of X
and Y, when X and Y are independent random variables.
4. For moment generating function of random variable X, prove that:
MaX (t) = MX (at),
MX+a (t) = eat MX (t).
5. Find the covariance of random variables X and Y having the joint probability density
function
x + y : 0 < x < 1, 0 < y < 1
f (x, y) =
0 : otherwise
6. Suppose that X and Y are independent random variables with probability densities
8/x3 :x>2
g(x) =
0 : otherwise
2y :0<y<1
h(y) =
0 : otherwise
Find the expected value of Z = XY.
7. Let X represent the number that occurs when a red die is tossed and Y the number
that occurs when a green die is tossed. Find E[(X+Y)], E[(X-Y)], E[(XY)].
8. A random variable X has the discrete uniform distribution
1/k : x = 1, 2, ..., k
f (x; k) =
0 : otherwise
Find moment generating function of X.
9. A random variable X has the geometric distribution g(x; p) = pq x−1 for x = 1, 2,
3,... Find moment-generating function of X and then use MX (t) to find the mean
and variance of the geometric distribution.
−λ x
10. A random variable X has the Poisson distribution p(x; λ) = e x!λ for x = 0, 1, 2,...
Find moment-generating function of X and then use MX (t) to find the mean and
variance of the Poisson distribution.
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