The joint probability mass function of the random variables X, Y, Z is

$$p ( 1,2,3 ) = p ( 2,1,1 ) = p ( 2,2,1 ) = p ( 2,3,2 ) = \frac { 1 } { 4 }$$

Find (a) E[XYZ], and (b) E[XY+XZ+YZ].

The joint probability mass function of the random variables X, Y, Z is

$$p(1,2,3)=p(2,1,1)=p(2,2,1)=p(2,3,2)=\frac{1}{4}$$

Find (a) E[XYZ], and (b) E[XY + XZ + YZ].

The joint probability mass function of X and Y is given by

$$\begin{array} { l l } { p ( 1,1 ) = \frac { 1 } { 8 } } & { p ( 1,2 ) = \frac { 1 } { 4 } } \\ { p ( 2,1 ) = \frac { 1 } { 8 } } & { p ( 2,2 ) = \frac { 1 } { 2 } } \end{array}$$

(a) Compute the conditional mass function of X given Y=i, i=1, 2. (b) Are X and Y independent? (c) Compute P{XY

$$\leq 3 \} , P \{ X + Y > 2 \} P \{ X / Y > 1 \}$$

Let the random variables X and Y have the joint pmf f(x, y) = x + y/32, x = 1, 2, y = 1, 2, 3, 4. Find the means

$$μ_X$$

and

$$μ_Y$$

, the variances

$$σ^2_X$$

and

$$σ^2_Y$$

, and the correlation coefficient ρ.

Assume a curve is given by the parametric equations x=f(t) and y=g(t), where f and g are twice differentiable. Use the Chain Rule to show that $y ^ { \prime \prime } ( x ) = \frac { f ^ { \prime } ( t ) g ^ { \prime \prime } ( t ) - g ^ { \prime } ( t ) f ^ { \prime \prime } ( t ) } { \left( f ^ { \prime } ( t ) \right) ^ { 3 } }$.

Sketch the probability mass function of a binomial distribution with n = 10 and p = 0.01 and comment on the shape of the distribution. a. What value of X is most likely? b. What value of X is least likely?

Four equal charges of

$$5.0 \mu \mathrm { C }$$

are placed at the vertices of a square of side 10 cm. a. Calculate the value of the electric potential at the centre of the square. b. Determine the electric field at the centre of the square. c. How do you reconcile your answers to a and b with the fact that the electric field is the derivative of the potential?

The random variables X and Y have joint density function

$$f ( x , y ) = 12 x y ( 1 - x ) \quad 0 < x < 1,0 < y < 1$$

and equal to 0 otherwise. (a) Are X and Y independent? (b) Find E[X]. (c) Find E[Y]., (d) Find Var(X). (e) Find Var(Y).

Without looking at the electron affinity table, arrange the following elements in order of decreasing electron affinities: C, O, Li, Na, Rb, and F.

Each throw of an unfair die lands on each of the odd numbers 1, 3, 5 with probability C and on each of the even numbers with probability 2C. (a) Find C. (b) Suppose that the die is tossed. Let X equal 1 if the result is an even number, and let it be 0 otherwise. Also, let Y equal 1 if the result is a number greater than three and let it be 0 otherwise. Find the joint probability mass function of X and Y. Suppose now that 12 independent tosses of the die are made. (c) Find the probability that each of the six outcomes occurs exactly twice. (d) Find the probability that 4 of the outcomes are either one or two, 4 are either three or four, and 4 are either five or six. (e) Find the probability that at least 8 of the tosses land on even numbers.

Write and solve a proportion for each situation. A daffodil that is 1.2 feet tall casts a shadow that is 1.5 feet long. At the same time, a nearby lamppost casts a shadow that is 20 feet long. What is the height of the the lamppost?

Lane 1 at your local track is 0.25 mi long. You live 0.5 mi away from the track. Which of the following results in the shortest jog? F. jogging 6 times around the track in Lane 1 G. jogging to the track and then 5 times around the tracking Lane 1 H. jogging to the track, 3 times around the track in Lane 1, and then home I. jogging 8 times around the track in Lane 1

Identify whether the equation is open or closed. If it is closed, identify if it is true or false. Then explain why. 63 - 8 = 55

Let the joint pmf of X and Y be f(x, y) = 1/4, (x, y) ∈ S = [(0, 0), (1, 1), (1, -1), (2, 0)]. a. Are X and Y independent? b. Calculate Cov(X, Y) and ρ. This exercise also illustrates the fact that dependent random variables can have a correlation coefficient of zero.