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Helen believes that the random variable $C$, representing cloud cover from the large data set, can be modeled by a discrete uniform distribution - Edexcel - A-Level Maths Mechanics - Question 1 - 2018 - Paper 2
Question 1
Helen believes that the random variable $C$, representing cloud cover from the large data set, can be modeled by a discrete uniform distribution. (a) Write down the... show full transcript
Worked Solution & Example Answer:Helen believes that the random variable $C$, representing cloud cover from the large data set, can be modeled by a discrete uniform distribution - Edexcel - A-Level Maths Mechanics - Question 1 - 2018 - Paper 2
Step 1
Write down the probability distribution for $C$
Answer
The random variable is modeled by a discrete uniform distribution, which can be defined as follows:
For a discrete uniform distribution where can take values , the probability mass function is given by:
P(C = c_i) = rac{1}{n} for .
In this case, if we assume can take values from to , then the probability for each value is given by:
P(C = c) = rac{1}{n+1} where .
Step 2
Using this model, find the probability that cloud cover is less than 50%
Answer
To find the probability that the cloud cover is less than 50%, we need to consider the range of values that can take.
If we assume the maximum value b of is (representing 100% cloud cover), then the values less than are .
Thus, we have a total of favorable outcomes. The total number of possible outcomes is (from to ).
Therefore, the probability that cloud cover is less than 50% is:
P(C < 50) = rac{50}{101} \approx 0.495
Step 3
Comment on the suitability of Helen's model in the light of this information
Answer
Given that Helen's model predicts a distribution of cloud cover that aligns with a uniform probability, but data from 2015 reveals that only of the days had cloud cover of less than %, it indicates that Helen's model may not accurately represent the true distribution of cloud cover. The model suggests a higher occurrence of lower cloud cover than observed in actual data, revealing a potential discrepancy.
This discrepancy calls into question the reliability of a discrete uniform model in this context, suggesting that a different probability distribution might be more suitable for representing the data.
Step 4
Suggest an appropriate refinement to Helen’s model
Answer
To refine Helen's model, one option may be to use a different probability distribution that accommodates the observed data more accurately. A possible refinement could involve:
- Utilizing a binomial distribution or a normal distribution if the data exhibits a normal tendency around a mean, allowing for more realistic probabilities that reflect the observed occurrence of cloud cover.
- Introducing weighting factors or adjusting the discrete uniform distribution to prioritize lower probabilities for greater cloud cover values based on historical data, which would be more representative of the reality observed.