• Probability distributions describe how the probabilities of different outcomes are distributed for a random variable, defining the odds of each possible value for the variable.
• In the following slides we will explore the differences between Continuous and Discrete Probability Distributions.
• We will also showcase some examples with plots created in R.

• A distribution is considered discrete if the random variable can only take on a finite set of values.
• Uses the Probability Mass Function (PMF) to determine the probabilities for each potential value.
• Total probability cannot exceed 1, so the sum of all probabilities is equal to 1.
• Example: Binomial Distribution.

\[ P(X = x) = \binom{n}{x}p^{x}(1-p)^{n-x} \]

• A distribution is considered continuous if the random variable can take on any value within a provided range or interval.
• Uses the Probability Density Function (PDF) to determine the probabilities for each potential value.
• Total probability cannot exceed 1, so the sum of all probabilities is equal to 1.
• Example: Normal Distribution.

\[ P(a \leq X \leq b) = \int_a^b f(x) \, dx \]

binom_dist_data <- rbinom(500, size = 25, prob = 0.5) 
#generate random binom dist.

ggplot(data.frame(binom_dist_data), aes(x = binom_dist_data)) +
  geom_histogram(binwidth = 1, fill = "red", color = "black") +
  labs(
    title = "Binomial Distribution Histogram",
    x = "Number of Successes",
    y = "Frequency"
  ) +
  theme(
    plot.title = element_text(hjust = 0.5)
  )
#plot histogram with red-filled bars and black outline
#labeled both x and y axis.

• Coin Flip Example
  • Probability of Success: 50%
  • 500 random events
  • 25 total trials

• Reaction Speed Example
  • Mean Reaction Speed: 25ms
  • Standard Deviation: 5ms
  • Participant Sample Size: 500

• Younger/Older Gamers with a mean of 25/35(ms) and standard deviation of 5/7(ms).