FIT2086 Assignment 1

Question 1

Given the team lost the previous game, the chance they win the next one is about 71.3%

Given a previous win, the team’s probability of winning the next game is 50%.

Using the table values we obtain and , while the marginal win probability is . Because these conditional probabilities differ from each other and from the marginal, the previous result does affect the chance of winning the current game.

Given the stated conditions, the probability the team wins exactly one of the next two games is about 41.6%.

Question 2

Thus the pmf is at the four endpoints and elsewhere on (and 0 outside)

Using the second-order Taylor (delta) method around with g(s)=1/s, we compute and plug in . Then we get

Because are independent, .
After computing the means and variances, we get and . Hence

Question 3

Let’s calculate the intersection of these two
Plot using R code

Integrate over [0,1]. Since and a>0, the power-rule integral applies. Simplifying gives .

Question 4

1. Let X be the number of α–particles in 10 seconds from 1 ng. i.e.
And we want
Thus, use the complement we get:
Choose to use the R to calculate this,



• Thus we get the answer, 0.3212264.
 

2. The emissions follow a Poisson model. The rate for 10 seconds is 8.3, so for 60 seconds:
We want:
Choose R to calculate this,



• Thus we get the answer, 0.09045661.

3. Count in each 10-second interval so . We know disjoint intervals are independent. Then,
Over five independent intervals, the number of successes is and we need Now we use the R to calculate,



Thus,
The probability is 57.36%

4. Use R to figure out

Hence we conclude the new material is from isotope A.