Cumulative Distribution Function (CDF) 2/2
Probability mass function over a range:
In some cases, we want to know the probability that a random variable is equal to numbers over a range.
Formula:
P(a < X <= b) = Fx(a) - Fx(b)
- P(a < X <= b) - the probability that a random variable X takes a value within the rage (a; b].
- Fx(a) - applying CMT to find a probability that a random variable X takes a value less than or a.
- Fx(b) - applying CMT to find a probability that a random variable X takes a value less than or b.
Example:
Calculate the probability a fair coin will succed in no more than 8 but no less than 4 cases (4; 8] if we have 15 attempts. We assume that success means getting a head.
Python realization:
12345678910111213141516171819# Import required library import scipy.stats as stats # The probability of getting 8 successes prob_8 = stats.binom.pmf(8, n = 15, p = 0.5) # The probability of getting 4 success prob_4 = stats.binom.pmf(4, n = 15, p = 0.5) # The resulting probability probability = prob_8 - prob_4 print("The probability is", probability * 100, "%")
Explanation
According to the formula, we subtract the probability that a random variable will take a value less than or four from the probability that a random value will take a value less than or 8.
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Probability mass function over a range:
In some cases, we want to know the probability that a random variable is equal to numbers over a range.
Formula:
P(a < X <= b) = Fx(a) - Fx(b)
- P(a < X <= b) - the probability that a random variable X takes a value within the rage (a; b].
- Fx(a) - applying CMT to find a probability that a random variable X takes a value less than or a.
- Fx(b) - applying CMT to find a probability that a random variable X takes a value less than or b.
Example:
Calculate the probability a fair coin will succed in no more than 8 but no less than 4 cases (4; 8] if we have 15 attempts. We assume that success means getting a head.
Python realization:
12345678910111213141516171819# Import required library import scipy.stats as stats # The probability of getting 8 successes prob_8 = stats.binom.pmf(8, n = 15, p = 0.5) # The probability of getting 4 success prob_4 = stats.binom.pmf(4, n = 15, p = 0.5) # The resulting probability probability = prob_8 - prob_4 print("The probability is", probability * 100, "%")
Explanation
According to the formula, we subtract the probability that a random variable will take a value less than or four from the probability that a random value will take a value less than or 8.
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