Use the applet found at (works with Internet Explorer only) to compute the proba
ID: 3181029 • Letter: U
Question
Use the applet found at
(works with Internet Explorer only) to compute the probabilities that one month later (30 days later), the weather is classified as sunny, cloudy or rainy
1. On any given day, the weather in Melbourne can be classified as sunny, cloudy or rainy. Assume that there can never be two sunny days in a row and that any sunny day is equally likely to be followed by a cloudy or rainy day, that cloudy and rainy days have a 50% chance of being followed by the same weather on the next day and that if there is change from cloudy or rainy, it is equally likely to be to a sunny day. (a) Write the probability transition matrix of this Markov chain. (b) On a Monday of a given week, the weather is cloudy. What is the probability that it is sunny on Tuesday? (c) What is the probability that it is sunny on Thursday? (d) What is the probability that the weather is classified as cloudy for all seven days of that week?Explanation / Answer
(a) From the information given in the problem statement, the TPM is given by
(b) If the Monday of a given week is cloudy, the probability that it is sunny on Tuesday, as seen from the above TPM, is 0.25.
(c) To obtain the probability of sunny on Thursday, starting Monday, we need to calculate the 3-step TPM which is done using R.
> Melbourne <- read.csv("clipboard",sep=" ",header=TRUE)
> Melbourne
Sunny Cloudy Rainy
1 0.00 0.50 0.50
2 0.25 0.50 0.25
3 0.25 0.25 0.50
> library(ACSWR)
> round(msteptpm(as.matrix(Melbourne),3),4)
Sunny Cloudy Rainy
[1,] 0.1875 0.4062 0.4062
[2,] 0.2031 0.4062 0.3906
[3,] 0.2031 0.3906 0.4062
If Monday is cloudy, the probability of Sunny on Thursday is 0.2031.
(d) The probability that the weather is classified as cloudy for all seven days of that week is provided by the 7 step TPM, which is now given by:
> round(msteptpm(as.matrix(Melbourne),7),4)
Sunny Cloudy Rainy
[1,] 0.2 0.4 0.4
[2,] 0.2 0.4 0.4
[3,] 0.2 0.4 0.4
Thus, the probability is 0.4.
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