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Please enable JavaScript in your browser settings or update your browser.Cumulative Distribution Function (CDF) 2/2 | Probability Functions
Probability Theory Update
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Conteúdo do Curso

Probability Theory Update

Probability Theory Update

1. Probability Basics
Set TheoryProbability ExperimentStatistical Independence
2. Statistical Dependence
Dependent EventsMutually Exclusive EventsAddition Rule for Mutually Exclusive EventsAddition Rule for Non-Mutually Exclusive EventsMultiplication Rule for Independent EventsMultiplication Rule for Dependent EventsConditional probabilityBayes Rule
3. Learn Crucial Terms
Random VariableDiscrete vs Continuous Random VariableExpected ValueThe Law of Large NumbersCMT
4. Probability Functions
Binomial DistributionProbability Mass Function (PMF) 1/2Probability Mass Function (PMF) 2/2Cumulative Distribution Function (CDF) 1/2Cumulative Distribution Function (CDF) 2/2Probability Density Function (PDF) 1/2Probability Density Function (PDF) 2/2
5. Distributions
Poisson Distribution 1/3Poisson Distribution 2/3Poisson Distribution 3/3Standard Normal Distribution (Gaussian distribution) 1/2

bookCumulative 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, "%")
copy

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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Seção 4. Capítulo 5
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bookCumulative 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, "%")
copy

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.

Switch to desktopMude para o desktop para praticar no mundo realContinue de onde você está usando uma das opções abaixo
Tudo estava claro?

Como podemos melhorá-lo?

Obrigado pelo seu feedback!

Seção 4. Capítulo 5
toggle bottom row

bookCumulative 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, "%")
copy

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.

Switch to desktopMude para o desktop para praticar no mundo realContinue de onde você está usando uma das opções abaixo
Tudo estava claro?

Como podemos melhorá-lo?

Obrigado pelo seu feedback!

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, "%")
copy

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.

Switch to desktopMude para o desktop para praticar no mundo realContinue de onde você está usando uma das opções abaixo
Seção 4. Capítulo 5
Switch to desktopMude para o desktop para praticar no mundo realContinue de onde você está usando uma das opções abaixo
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