Question

In: Statistics and Probability

A fair 4-sided die is rolled, let X denote the outcome. After that, if X =...

A fair 4-sided die is rolled, let X denote the outcome. After that, if X = x, then x fair coins are tossed, let Y denote the number of Tails observed. a) Find P( X >= 3 | Y = 0 ). b) Find E( X | Y = 2 ). “Hint”: Construct the joint probability distribution for ( X, Y ) first. Write it in the form of a rectangular array with x = 1, 2, 3, 4 and y = 0, 1, 2, 3, 4.

Solutions

Expert Solution

a) The value of Y given different values of X here is computed as:

P(Y = 0 | X = 1) = 0.5
P(Y = 0 | X = 2) = 0.52 = 0.25
P(Y = 0 | X = 3) = 0.53 = 0.125
P(Y = 0 | X = 4) = 0.54 = 0.0625

Also for a fair 4 sided die, we get:
P(X = 1) = P(X = 2) = P(X = 3) = P(X = 4) = 0.25

Therefore, using law of total probability, we get here:
P(Y = 0) = P(Y = 0 |X = 1)P(X = 1) + P(Y = 0 | X = 2)P(X = 2) + P(Y = 0 | X = 3)P(X = 3) + P(Y = 0 | X = 4)P(X = 4)
P(Y = 0) = 0.25*(0.5 + 0.25 + 0.125 + 0.0625) = 0.234375

Now P(X >= 3 | Y = 0) is computed using Bayes theorem here as:
P(X >= 3 | Y = 0) = P(Y = 0 | X = 3)P(X = 3) + P(Y = 0 | X = 4)P(X = 4) / P(Y = 0)

= 0.25*(0.125 + 0.0625) / 0.234375

= 0.2

Therefore 0.2 is the required probability here

b) The value of Y given different values of X here is computed as:

P(Y = 2 | X = 1) = 0
P(Y = 2 | X = 2) = 0.52 = 0.25
P(Y = 2 | X = 3) = 3*0.53 = 0.375
P(Y = 2 | X = 4) = (4c2)*0.54 = 0.375

The expected value here is computed as:

E(X | Y = 2) = 0*P(Y = 2 | X = 1) + 2*P(Y = 2 | X = 2) + 3*P(Y = 2 | X = 3) + 4*P(Y = 2 | X = 4)

= 0*1 + 0.25*2 + 0.375*3 + 0.375*4

= 3.225

Therefore 3.225 is the expected value of X given Y = 2 here.


Related Solutions

Let X equal the outcome (1, 2 , 3 or 4) when a fair four-sided die...
Let X equal the outcome (1, 2 , 3 or 4) when a fair four-sided die is rolled; let Y equal the outcome (1, 2, 3, 4, 5 or 6) when a fair six-sided die is rolled. Let W=X+Y. a. What is the pdf of W? b What is E(W)?
Consider the experiment of rolling a six-sided fair die. Let X denote the number of rolls...
Consider the experiment of rolling a six-sided fair die. Let X denote the number of rolls it takes to obtain the first 5, Y denote the number of rolls until the first 2, and Z denote the number of rolls until the first 4. Numerical answers are needed only for parts (a) and (b). Expressions are sufficient for parts (c), (d), and (e). a) E[X|Y = 1 or Z = 1] b) E[X|Y = 1 and Z = 2] c)...
A fair six-sided die is rolled repeatedly until the third time a 6 is rolled. Let...
A fair six-sided die is rolled repeatedly until the third time a 6 is rolled. Let X denote the number of rolls required until the third 6 is rolled. Find the probability that fewer than 5 rolls will be required to roll a 6 three times.
Two fair dice are rolled at once. Let x denote the difference in the number of...
Two fair dice are rolled at once. Let x denote the difference in the number of dots that appear on the top faces of the two dice. For example, if a 1 and a 5 are rolled, the difference is 5−1=4, so x=4. If two sixes are rolled, 6−6=0, so x=0. Construct the probability distribution for x. Arrange x in increasing order and write the probabilities P(x) as simplified fractions.
Suppose two fair dice are rolled. Let X denote the product of the values on the...
Suppose two fair dice are rolled. Let X denote the product of the values on the dice and Y denote minimum of the two dice. Find E[X] and E[Y] Find Var X and Var Y Let Z=XY. Find E[Z]. Find Cov(X,Y) and Corr(X,Y) Find E[X|Y=1] and E[Y|X=1]
a die is rolled 6 times let X denote the number of 2's that appear on...
a die is rolled 6 times let X denote the number of 2's that appear on the die. 1. show that X is binomial. 2. what is the porbaility of getting at least one 2. 3. find the mean and the standard deviaion of X
A fair die is rolled twice. Let X and Y be the smallest and largest, respectively,...
A fair die is rolled twice. Let X and Y be the smallest and largest, respectively, number that appears in the two rolls. (a) Determine the probability mass function of (X, Y). (Write a formula forP(X=i, Y=j)or give a table of values.) (b) Are X and Y independent? (c) Find E(X+Y). (Give your answer as a decimal number.)
A fair die is rolled twice. Let X be the maximum of the two rolls. Find...
A fair die is rolled twice. Let X be the maximum of the two rolls. Find the distribution of X. Let Y be the minimum of the two rolls. Find the variance of Y.
Let X be the outcome of rolling a fair six-sided dice. The possible outcomes or X...
Let X be the outcome of rolling a fair six-sided dice. The possible outcomes or X are 1,2,3,4,5 and 6 and all are equally likely. What is the cumulative distribution function F(x)?
Example 4: A fair six-sided die is rolled six times. If the face numbered k is...
Example 4: A fair six-sided die is rolled six times. If the face numbered k is the outcome on roll k for k = 1, 2, 3, 4, 5, 6 we say that a match has occurred. The experiment is called a success if at least one match occurs during the six trials. Otherwise, the experiment is called a failure. The outcome space is O = {success, failure}. Let event A = {success}. Which value has P(A)? **This question has...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT