Three fair coins are flipped simultaneously. All the coins that come up tails are flipped once more. Let X denote the number of coins that come up heads on the first toss, and let Y represent the number of re-tossed coins that come up heads on the second flipping. (a) 0.5 pt - Find the admissible values for X, and find the possible values that Y can take conditioned on X. (b) 0.5 pt - Find the marginal PMF and the mean of X. (c) 1 pt - Find the joint PMF of X and Y. (d) 1 pt - Calculate the expected value of X+Y (possibly using indicator functions). (e) 1 pt - Derive a formula for E[Y|X] and use it to compute E[E[X +Y|X]]. (f) 1 pt - Find the PMF of U = X+Y and compute E[U].

Asmall (but heavy) particle placed in a glass of water will follow a zigzag motion because the particle will bounce off of the water molecules it meets. this is called brownian motion. a physicist simulates this on a computer, by varying the distance a particle can travel (called the mean free length), on average, before it collides with a water molecule and assigning the change in motion to be one of 8 directions, each with a similar probability. by running the simulated particle (with the same mean free length) many times she determines that it should take 15 seconds, on average, for the particle to fall to the bottom, with a standard deviation of 1.5 seconds. next she lets a real particle fall through a glass of water and finds that it took 18 seconds. what does she conclude, and why?

Ke’ajah has a coupon for 1/3 off the regular price, t, of a tent. which expression represents the price of the tent using the coupon? select the two correct expressions. a. 1/3t b. 2/3t c. t - 1/3 d. t - 2/3 e. t - 1/3t f. t - 2/3t

In the diagram, gef and hef are congruent. what is the value of x