Random variables describe the outcomes of a random process or experiment by assigning a numerical value to each possible outcome, allowing for a quantitative analysis of uncertain events.

A discrete random variable can take on countable number of distinct values. For example, heads or tail, the number of students in the class.

Continuous random variables can take on any value in interval. For example, pressure, height, mass and distance.

The expected value (aka mean) is the measure of the central tendency of a random variable.

The expected value represents the long-term average or the average value you would expect to obtain if you repeated an experiment or process a large number of times.

Formula: $E[X] = \sum_i x_i\cdot P(X=x_i)$

• $E $: Expected value
• $X$: Random variable
• $x_i$: Each possible value that the random variable $X$ can take on
• $P(X=x_i)$: Probability mass function (PMF) that assigns the probability of a random variable X taking on the specific value $x_i$

Both formulas involve summing up the product of each value and its corresponding weight or frequency.

• In the case of the expected value, the weights are the probabilities assigned by the probability mass function (PMF).
• In the case of the mean, the weights are the same for each value, and thus able to divide by the total number of observations $(\frac{1}{n})$.

Formula: $Var(X) = \sum_i(x_i-E[X])^2\cdot P(X=x_i)$

• $Var(X)$: Variance
• $E $: Expected value
• $X:$ Random variable
• $x_i$: Each possible value that the random variable $X$ can take on
• $P(X=x_i)$: Probability mass function (PMF) that assigns the probability of a random variable X taking on the specific value $x_i$

The mode of a discrete random variable is the value that occurs with the highest probability.

A probability distribution describes all the random variable’s possible outcomes (values) and the associated probabilities.

1. Non-negativity.
2. Each probability is between 0 and 1.
3. The sum of all the probabilities is 1.

• Definition: Describes the probability distribution over a discrete random variable (RV). It is a function that returns the probability of a RV being exactly equal to a specific outcome or value.
• Example(s): Bernoulli Distribution, Binomial Distribution, Poisson distribution

• Definition: Describes the probability distribution over a continuous random variable (RV). Where the random variable can take on any value within a certain range.
• Example(s): Normal Distribution, Exponential Distribution

The total area under for both continuous and discrete probability distribution curve is 1.

No, because the probability is associated with intervals rather than individual points.

CDF is the probability that a random variable $X$ will take a value less than or equal to $x$. It provides a cumulative view of the probability distribution.

The Bernoulli distribution is a type of discrete probability distribution that describes the probability of a single event happening (success) or not happening (failure). It has two parameters: the probability of success $(p)$ and the probability of failure $(q=1-p)$

• Flipping a coin (heads or tail)
• Passing an exam (pass of fail)

$P(k;p) = p^k(1-p)^{1-k}$

Where:

• $k$ is the possible outcomes (0 for failure, 1 for success)
• $p$ is the probability of success

$E(X)= 1\times p + 0\times(1-p) = p$

$Var(X)=E(X^2)−E(X)^2$

$E(X^2) = 1^2×p+0^2×(1−p) = p$

$E(X)^2 = p^2$

$Var(X) = p−p^2=p(1−p).$

The Binomial distribution is a type of discrete probability distribution that describes the probability of getting a certain number of successes in a fixed number of independent and identical Bernoulli trials, where each trial has two possible outcomes: success or failure.

• Counting the number of heads in a series of coin flips.
• Finding the number of students that will pass a series of independent exams.

1. Binary Outcomes: Each trial results in one of two possible outcomes, often referred to as success and failure.
2. Fixed Number of Trials ($n$): The number of trials is predetermined and remains constant throughout the experiment.
3. Independence: The outcome of one trial does not affect the outcome of another. Each trial is considered independent.
4. Constant Probability of Success ($p$): The probability of success (denoted by $p$) remains the same for each trial.

$P(X=k) = \binom nk p^k(1-p)^{1-k}$

Where:

• $k$ is the number of successes
• $n$ is the number of independent trials
• $p$ is the probability of success
• $\binom n k$ is the binomial coefficient, representing the number of ways to choose $k$ successes from $n$ trials.

The expected value of a binomially distributed random variable $X$ is the sum of $n$ identical Bernoulli random variables, each with expected value $p$, which gives us ..

$E(X)= np$

Similar the the expected value, the variance of a binomially distributed random variable $X$ is the sum of independent Bernoulli random variable variance $p(1-p)$, which gives us

$Var(X)= npq =np(1-p)$

The distribution shape approaches normality: According to the central limit theorem, as the number of trials becomes large, the shape of the Binomial distribution approaches that of a normal distribution.

Normal distribution (AKA Gaussian distribution) is a continuous probability distribution that is symmetric and bell-shaped, which implies that the frequency of observing datapoints is higher when it is nearer to the mean.

Many natural processes and measurements in the natural and social sciences such as heights, weights, IQ scores, are approximately normally distributed. For this reason, we are able to perform inferential statistics on them.

The mean, medan and mode are equal to one another and these values all represent the center of the distribution, which is also the peak.

The skewness measures the degree of symmetry of a distribution. The normal distribution is symmetric and thus, has a skewness = 0

The 68-95-99.7 rule, also known as the empirical rule is used to estimate the spread of data in a normal distribution.

• 68% of the data falls within 1 standard deviation of the mean.
• 95% of the data falls within 2 standard deviation of the mean.
• 99.7% of the data falls within 3 standard deviation of the mean.

The standard normal distribution (Z distribution) is a specific normal distribution with a mean of 0 and a standard deviation of 1.

$Z= \frac{x-\mu}{\sigma}$

• $Z$: Z-score
• $x$: Individual observation
• $\mu$: Mean
• $\sigma$: Standard deviation

Formula: $Z= \frac{x-\mu}{\sigma}$

• Shifting to the mean $(x-\mu)$:
  • Subtracting the mean shifts the distribution so that the mean becomes 0.
• Scaling the standard deviation $(\frac{1}{\sigma})$:
  • When we divide each data point by the standard deviation, we are expressing the data points in terms of unit of standard deviation from the mean. Hence the standard deviation is 1.

$Z$ score is a measure of how many standard deviations a particular data point is from the mean. The higher the $Z$ score, the further away the value is from the mean.

1. Positive $Z$ score indicates that the data point is above the mean of the distribution.
2. Negative $Z$ score indicates that the data point is below the mean of the distribution.

• Comparisons and Standardisation:
  • Standardising the data allows for meaningful comparisons across different normal distributions. It simplifies the analysis by providing a common scale for measurements, regardless of the original distribution’s parameters.
• Probability Calculations:
  • The transformation allow us to calculate probabilities associated with specific values in the standard normal distribution.
• Statistical Testing:
  • Many statistical tests and hypothesis testing procedures assume a standard normal distribution.

1. State the problem:
   • Finding the probability greater than a value, less than a value or within an interval.
2. Standardise the value:
   • Calculate the Z-score.
3. Determine the probability:
   • Find the corresponding cumulative probability of that $Z$ score.

• Formula: $P(Z\le a)$
• Explanation: The Z-score table have provided the cumulative probability which represents the probability that a randomly selected value from a standard normal distribution is less than or equal to $a$.

• Formula: $P(Z>a) =1- P(Z\le a)$
• Explanation: Subtracting $P(Z\le a)$ from 1 gives the probability that Z is greater than a.

• Formula: $P(a \le Z \le b) = P(Z \le b)-P(Z \le a)$
• Explanation: Subtracting $P(Z\le a)$ from $P(Z\le b)$ gives the probability of $Z$ falling within the interval $[a,b]$. We are removing the portion of the distribution less than a, leaving the portion between $a$ and $b$.