Let U have a U(0,1) distribution. describe how to simulate the outcome of a roll with a die using U. Define Y as follows: round 6U + 1 down to the nearest integer. What the popssible outcomes of Y and their probabilities?
Let U have a U(0,1) distribution. describe how to simulate the outcome of a roll with...
Use a random number generator to simulate the roll of a fair die 100 times. Let the number face up on the die represent the variable X A. Build a relative frequency table of the outcomes of the variable X. X Freq Rel. Freq B. Use the relative frequency distribution from part c to estimate the probability of an even number face up, then find the actual probability using the probability distribution and comment on the difference in values.
Let X have a U[0,1] distribution and Y have a Exp[1] distribution, what is the maximum expected value?
Describe how random numbers can be used to simulate the roll of a die. How can two dice be simulated using this function? How can three dice be simulated using this function?
Let U ~uniform(0,1). Let Y =−ln(1−U). hint: If FX (x) = FY (y) and supports x,y ∈ D, X and Y have the same distribution. Find FY (y) and fY (y). Now, it should be straight forward that Y follows distribution with parameter_____________-
U is Uniform distribution here
Let X ~ U[0,1] and Y = max {,x) (a) Is Y a continuous random variable? Justify (b) Compute E[Y]. (Hint: Note that when a (Hint: Note that when a-, max 1.a- , and when a > ļ, max | , a- ax {3a, and when a > a
5. Let X have a uniform distribution on the interval (0,1). Given X = x, let Y have a uniform distribution on (0, 2). (a) The conditional pdf of Y, given that X = x, is fyıx(ylx) = 1 for 0 < y < x, since Y|X ~U(0, X). Show that the mean of this (conditional) distribution is E(Y|X) = , and hence, show that Ex{E(Y|X)} = i. (Hint: what is the mean of ?) (b) Noting that fr\x(y|x) =...
Let the random variable X have a uniform distribution on [0,1] and the random variable Y (independent of X) have a uniform distribution on [0,2]. Find P[XY<1].
(c) (20 pts.) Let X have a uniform distribution U(0, 2) and let the considiton; distribution of Y given X = x be U(0, x3) i. Determine f (x, y). Make sure to describe the support of f. ii. Calculate fy (y) iii. Find E(Y).
Suppose you roll two 4 sided dice. Let the probabilities of the first die be represented by random variable X and those of the second die be represented by random variable Y. Let random variable Z be |X-Y|. The range of Z will be from 0 to 3. 1. Find E(Z). 2. Find P(Z = 1|Z <3). 3. What is the probability that you have to play the game 4 times before you roll Z=3? 4. What is the probability...
Suppose you roll two 4 sided dice. Let the probabilities of the first die be represented by random variable X and those of the second die be represented by random variable Y. Let random variable Z be X-Y). The range of Z will be from 0 to 3. 1. (2.5 points) Find E(Z). 2. (2.5 points) Find P(Z = 21Z >1). 3. (2.5 points) What is the probability that you have to play the game 6 times before you roll...