Section 6.1 & 6.2

Random variable: A random variable takes numerical values that describe

the outcomes of a random process.

Probability distribution: The probability distribution of a random variable gives

its possible values and their probabilities.

Discrete random variable: A discrete random variable X takes a fixed set of

possible values with gaps between them.

Mean (expected value) of a discrete random variable: The mean (expected value) of a discrete random variable is its average value over many, many trials of the same random process. Suppose that X is a discrete random variable with probability distribution:

To find the mean (expected value) of X, multiply each possible value of X by its probability, then add all the products:

X = E(X) = x1p1+ x2p2 + x3p3+... = XiPi

Standard deviation of a discrete random variable: The standard deviation of a discrete random variable measures how much the values of the variable typically vary from the mean in many, many trials of the random process. Suppose that X is a discrete random variable with probability distribution

and that μx is the mean of X.

Variance: The variance of X is

The standard deviation of X is the square root of the variance:

Continuous random variable: A continuous random variable can take any value in an interval on the number line.

  • Adding a positive constant a to (subtracting a from) a random variable increases (decreases) measures of center and location by a, but does not affect measures of variability (range, IQR, standard deviation) or the shape of its probability distribution.

  • Multiplying (dividing) a random variable by a positive constant b multiplies (divides) measures of center and location by b and multiplies (divides) measures of variability (range, IQR, standard deviation) by b, but does not change the shape of its probability distribution.

  • If Y = a + bX is a linear transformation of the random variable X with b > 0,

    • The probability distribution of Y has the same shape as the probability distribution of X.

    • μY = a + bμX

    • σY = bσX

  • If X and Y are any two random variables,

    • μX+Y = μX + μY : The mean of the sum of two random variables is the sum of their means.

    • μX−Y = μX − μY : The mean of the difference of two random variables is the difference of their means.

  • If X and Y are independent random variables, then knowing the value of one variable does not change the probability distribution of the other variable. In that case, variances add:

    • σ 2 X+Y = σ 2 X + σ 2 Y : The variance of the sum of two independent random variables is the sum of their variances. 

    • σ 2 X−Y = σ 2 X + σ 2 Y : The variance of the difference of two independent random variables is the sum of their variances. 

  • To get the standard deviation of the sum or difference of two independent random variables, calculate the variance and then take the square root:

    • σX+Y = σX−Y =σ2 X + σ2 Y

  • If aX + bY is a linear combination of the random variables X and Y,

    • Its mean is aμX + bμY .

    • Its standard deviation is a2σ2 X +b2 σ2 Y if X and Y are independent. 

  • A linear combination of independent Normal random variables is a Normal random variable.