Course Content

Probability Theory Update

1. Probability Basics

Set Theory Probability Experiment Statistical Independence

2. Statistical Dependence

Dependent Events Mutually Exclusive Events Addition Rule for Mutually Exclusive Events Addition Rule for Non-Mutually Exclusive Events Multiplication Rule for Independent Events Multiplication Rule for Dependent Events Conditional probability Bayes Rule

3. Learn Crucial Terms

Random Variable Discrete vs Continuous Random Variable Expected Value The Law of Large Numbers CMT

4. Probability Functions

Binomial Distribution Probability Mass Function (PMF) 1/2 Probability Mass Function (PMF) 2/2 Cumulative Distribution Function (CDF) 1/2 Cumulative Distribution Function (CDF) 2/2 Probability Density Function (PDF) 1/2 Probability Density Function (PDF) 2/2

5. Distributions

Poisson Distribution 1/3 Poisson Distribution 2/3 Poisson Distribution 3/3 Standard Normal Distribution (Gaussian distribution) 1/2

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.

Switch to desktop for real-world practiceContinue from where you are using one of the options below

Everything was clear?

Section 4. Chapter 5

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.

Switch to desktop for real-world practiceContinue from where you are using one of the options below

Everything was clear?

Section 4. Chapter 5

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.

Switch to desktop for real-world practiceContinue from where you are using one of the options below

Everything was clear?

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.

Switch to desktop for real-world practiceContinue from where you are using one of the options below

Section 4. Chapter 5

Switch to desktop for real-world practiceContinue from where you are using one of the options below