Let X be the number of heads in two tosses of a fair coin. Suppose a fair die is rolled X+1 times after the value of X is determined from the coin tosses. Let Y denote the total of the face values of the X+1 rolls of the die. Find E[Y | X = x] and V[Y | X = x] as expressions involving x. Use these conditional expected values to find E[Y] and V[Y].

Let ? ∈ {1, 2} and ? ∈ {3, 4} be independent random variables with PMF-s: ??(1)= 1/2 ??(2)= 1/2 ??(3)= 1/3 ??(4)= 2/3

Answer the following questions

(a) Write down the joint PMF

(b) Calculate?(?+?≤5)and?(? −?≥2) 2 ?2+1

(c) Calculate ?(?? ), ?(? ? ), E ? −2
(d) Calculate the C??(?, ? ), C??(1 − ?, 3? + 2) and V??(2? − ? )

(?*) Calculate C??(??, ?), C??(??, ? + ? ) and V?? ?

If a golf ball were dropped from a significant height and the force of Drag is modeled by Fd=0.0057*V-0.0106, where V is the velocity in m/s the Fd is the force of drag in N. The mass of the golf ball is 0.0455 Kg.

h. How high will the golf ball bounce after the 5th impact with the concrete surface ignore drag force?

i. If drag was included for G and H what type of analysis technique would be needed

A mass of 0.3 kg of saturated refrigerant-134a is contained in a piston-cylinder device at 240 kPa.
Initially, 70 percent of the mass is in the liquid phase. Now heat is transferred to the refrigerant at
constant pressure until the cylinder contains vapor only.
(a) show the process on a P-v and T-v diagrams with respect to saturation lines. Determine;
(b) the volume occupied by the refrigerant initially,
(c) the work done, and
(d) the total heat transfer.

1. A rechargeable battery with emf 1.52 V and internal resistance 0.75 Ω is connected positive to positive and negative to negative with a 3.00 V battery with internal resistance 1.4 Ω. Assume the emf of each battery remains constant.

(a) Find the current.

(b) How much energy will be stored in the rechargeable battery in 1.00 minute?

(c) Determine the efficiency of this charging arrangement.

(d) How could the efficiency be improved (and wasted energy be reduced)?

The figure shows two masses connected by a cord passing over a pulley of radius R₀ and moment of inertia I. Mass M₁slides on a frictionless surface, and M₂ hangs freely.

(a) Determine a formula for the angular momentum of the system about the pulley axis, as a function of the speed v of mass M₁ or M₂. (Use R0 for R₀, M1 for M₁, M2 for M₂, I,v and g for gravity, as necessary.)

(b) Determine a formula for the acceleration of the masses.

In a model AC generator, a 497 turn rectangular coil 14.0 cm by 23 cm rotates at 115 rev/min in a uniform magnetic field of 0.58 T.

(a) What is the maximum emf induced in the coil?
V

(b) What is the instantaneous value of the emf in the coil at t = (?/39) s? Assume that the emf is zero at t = 0.
V

(c) What is the smallest value of t for which the emf will have its maximum value?
s

two skaters collide and embrace, in a completely inelastic collision. boomer, whose mass is 83 kg, is originally moving east with v=6.2km/hr. sooner whose. mass is 55kg, is originally moving north with v=7.8km/hr. (a) what is the velocity of the couple after impact? (b) what is the velocity of the center of mass of two skaters before and aafter collision? (c) what is the fractional change in the Kinect energy of the skaters because of the collision?