Reference no: EM132312162
Assignment - Elements of Probability
Q1. Consider two random variables X and Y with the joint probability density
![2168_figure.png](https://secure.expertsmind.com/CMSImages/2168_figure.png)
Let Z = XY2 and W = Y be a joint transformation of (X, Y).
(a) Find the support of (Z, W).
(b) Find the inverse transformation.
(c) Find the Jacobian of the inverse transformation.
(d) Find the joint pdf of (Z, W).
(e) Find the pdf of Z = XY2 from the joint pdf of (Z, W).
Q2. Let X1, X2, · · · , Xn be a random sample of size n from a geometric distribution with pmf
f(x) = 0.75 · 0.25x-1, x =1, 2, 3, · · · .
(a) Find the mgf MY_3(t) of Y3 = X1 +X2 +X3 using the geometric mgf. Then name the distribution of Y3.
(b) Find the mgf MY_n(t) of Yn = X1 +X2 + · · · + Xn. Then name the distribution of Yn.
(c) Find the mgf MY¯_n(t) of the sample mean Y¯n = Yn/n.
(d) Find the limit limn→∞MY¯_n(t) using the result of (c). What distribution does the limiting mgf correspond to?
(e) Let Zn = 3/2√n Y¯n - 2√n. Find MZ_n(t), the mgf of Zn. Then ?nd the limiting mgf limn→∞MZ_n(t). What is the limiting distribution of Zn?
Q3. Consider two random variables X and Y with the joint probability density
![2039_figure1.png](https://secure.expertsmind.com/CMSImages/2039_figure1.png)
which is the same as in Question 1. Now let Z = XY2 and U = X be a joint transformation of (X, Y).
(a) Find the support of (Z, U).
(b) Find the inverse transformation.
(c) Find the Jacobian of the inverse transformation.
(d) Find the joint pdf of (Z, U).
(e) Find the pdf of Z = XY2 from the joint pdf of (Z, U).
Q4. Let X1, X2 and X3 be a random sample (i.e. X1, X2, X3 are independent) of size n =3 from the exponential distribution with pdf f(x)= ½e-x/2, 0 < x < ∞. Find
(a) P(1 <X1 < 2, 2 < X2 < 3, 3 < X3 < 4).
(b) E[X1X22(X3 - 2)2].
Q5. Let X1, X2, X3 denote a random sample of size n =3 from a distribution with the Poisson pmf
f(x)= (2x/x!)e-2, x = 0, 1, 2, 3 · · · .
(a) Compute P(X1 + X2 + X3 =1).
(b) Find the moment-generating function of Z = X1 +X2 +X3 using the Poisson mgf of X1. Then name the distribution of Z.
(c) Find the probability P(X1 + X2 + X3 =4) using the result of (b).
(d) If Y =max{X1, X2, X3}. Find the probability P(Y ≤ 2).
Q6. If E(X) = 16 and E(X2) = 292, use Chebyshev's inequality to determine
(a) A lower bound for P(8 < X < 24).
(b) An upper bound for P(|X - 16| ≥ 18).
Q7. Suppose that the distribution of the weight of a prepackaged "1-pound bag" of carrots is N(1.18, 0.072) and the distribution of the weight of a prepackaged "3-pound bag" of carrots is N(3.22, 0.092). Now independently select at random three 1-pound bags of carrots with weights being X1, X2 and X3 respectively. Also randomly select one 3-pound bag of carrots with weight being W. Let Y = X1 + X2 + X3.
(a) Find the mgf of Y.
(b) Find the distribution of Y, the total weight of the three 1-pound bags of carrots selected.
(c) Find the probability P(Y < W), i.e., the probability that the sum of weights of three 1-pound bags randomly selected is smaller than the weight of one 3-pound bag randomly selected.