Test Your Probability Knowledge

Random Variables and Probability Distributions

• Random Variable is a function that maps each sample point to a real number.
• Use uppercase letter to denote a random variable.
• Discrete Random Variables: finite number of possibilities or an unending sequence with as many elements as there are whole numbers.
• Continuous Random Variable: infinite number of possibilities equal to the number of points on a line segment.
• Probability Distributions: Cumulative distribution function (cdf), denoted by F(*) is defined by F(x)=P(X<=x).
• Discrete Probability Functions: a table or formula listing all possible values that a discrete random variable can take on along with the associated probabilities.
• Probability Mass Function: p(x) = P (X=x); mass points, values of X which p(x) > 0.
• Probability Density Functions: Continuous random variable X, which satisfies the following: f(x) >= 0 for all x, the total area under its curve = 1, and P(a <= X <= b) is the area bounded by the curve.
• Expected Value of X: Let X be a discrete random variable with probability mass function.
• Variance: Let X be the random variable with mean mu, then the variance of X is Var(X) = E[(X-mu)^2].
• Standard Deviation: the positive square root of variance; variance of X is a measure of dispersion.
• Binomial Experiment: possesses the following: the experiment consists of n identical trials, each trial results in one of 2 outcomes, a success or a failure, probability of a success on a single trial is equal to p and remains the same from the trial to trial, and trials are independent.
• Normal Distribution: a continuous random variable X is said to be normally distributed or follows normal distribution if its PDF is.