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Random Variables and Mean, Variance, and Standard Deviation Calculation, Study notes of Statistics

Ohio State University (OSU) - LimaStatistics

This document from a stat 528 (autumn 2008) lecture covers the concept of random variables, their mean, variance, and standard deviation. The difference between discrete and continuous random variables, and provides examples and calculations for both types. It also covers the rules for means and variances, and the relationship between the mean and standard deviation for normal and uniform distributions.

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Stat 528 (Autumn 2008) – Elly Kaizar
Random Variables
Reading: Sections 4.3, 4.4.
•Random variables (RVs)
–Discrete and continuous RVs
•The mean of a RV
–The mean of discrete RV
–The mean of some continuous RVs
–Transformations
–Rules for means
•The variance and standard deviation of a RV
–Rules for variances
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Download Random Variables and Mean, Variance, and Standard Deviation Calculation and more Study notes Statistics in PDF only on Docsity!

Stat 528 (Autumn 2008) – Elly Kaizar Random Variables

Reading: Sections 4.3, 4.4.

  • Random variables (RVs)
    • Discrete and continuous RVs
  • The mean of a RV
    • The mean of discrete RV
    • The mean of some continuous RVs
    • Transformations
    • Rules for means
  • The variance and standard deviation of a RV
    • Rules for variances

Random variables

  • Recall: “A variable is any characterisitc of an individual. A variable can take different values for different individuals.”
  • A random variable (RV), X, is a variable that depends on the outcome of a chance experiment or a random phe- nomenon. A random variable must have a numeric value.
  • We can have discrete or continuous RVs.
  • Examples:
    • Number of heads in three coin flips.
    • Height of students selected at random from the stat 528 class.
    • In a random sample of components, the number that pass a test.
    • Number of particles counted by a Geiger counter in a radiation experiment.

Telephone example

Suppose that the length X of an international telephone call, to the nearest minute, is given by

value of X 1 2 3 4 probability 0.2 0.5 0.2 0.

Calculate the following.

  1. P (X ≤ 2) =

2. P (X < 2) =

3. P (X ≥ 1) =

Random walk example

A fly leaves a restaurant. Every minute thereafter the fly ran- domly moves either 1 meter left (-1) with probability 0.5 or 1 meter right (+1) with probability 0.5. Let the RV X denote the distance the fly moves left or right in three minutes, relative to his start position. What is the probability distribution of X?

Continuous random variables

  • A continuous RV, X takes values x anywhere in an inter- val of values. This interval could be unbounded = (−∞, ∞).
  • Example: Consider the direction of a spinner. What is the probability that the spinner lands between 90o^ and 180o?
  • Probability distributions for continuous RVs are de- scribed by the probability density curve. 1. The density curve always has non-negative height. 2. The area under the density curve is one. (Compare with the probability distribution for discrete RVs).

Calculating probabilities

  • Probabilities are given by the area under the curve. e.g. P (a < X < b) is
  • For any one value x of X, P (X = x) = 0,

because there is no area at one point. But, the height under the probability density curve need not be zero.

The normal distribution

  • See the previous notes!
  • Note: When answering questions using the normal probabil- ity distribution we should be careful to phrase our answers carefully. e.g., P (a ≤ X ≤ b) rather than a ≤ X ≤ b.

SRS example

An opinion poll asks a SRS of 1500 adults, “do you happen to jog?” Suppose that in fact 15% of adults would answer yes to this question. However the proportion, p̂ , of the sample who answer “yes” in this sample will vary in repeated sampling. We will show later in this class that we can suppose that p̂ is normally distributed with mean μ = 0.15 and standard deviation σ = 0 .0092. Find the probability that either less than 14% or over 16% of the polled adults claim to jog.

The mean or expected value of a discrete RV

  • Suppose we have a discrete RV with probability distribution: value of X x 1 x 2 x 3... xk probability p 1 p 2 p 3... pk
  • Then, the mean or expected value of X is

μX =

∑^ k i=

xi pi.

  • Intuition: Suppose pi = p for all i = 1,... , k.

pi = p =

μX = ∑ki=1 xi pi =

Discrete Mean Example

  • Ex: Remember that the length X of an international tele- phone call, to the nearest minute, is given by

value of X 1 2 3 4 probability 0.2 0.5 0.2 0. What is the mean length of an international telephone call?

The mean of a continuous RV

  • Harder to calculate (need calculus!). FYI: μY = ∫^ x f (x) ∂x, f (x) = probability density function.
  • The normal and uniform distributions are both symmetric. In these cases the mean is equal to the median. Thus: - The mean of a N(μ, σ) RV is μ. - The mean of a U(a, b) RV is (a + b)/2. (draw the picture for each case).

Rules for means

  • Let X and Y be discrete or continuous RVs. Then μX+Y = μX + μY.
  • Let a and b be fixed numbers. Then μa+bX = a + bμX.

(note this is a linear transformation)

  • Example: Look again at the telephone example, where we calculate μY , where Y = 8X + 5.

The variance and stdev of certain continuous RVs

  • You should know that:
    • The variance of a N(μ, σ) RV is σ^2 and the stdev is σ.
    • The variance of a U(a, b) RV is (b − a)^2 /12 and so the standard deviation is |b − a|/√12.

Rules for variances

  • Let X and Y be discrete or continuous RVs, and let a and b be fixed numbers. Then σ a^2 +bX = b^2 σ^2 X.
  • IF X and Y are independent then σ X^2 +Y = σ^2 X + σ Y^2.
  • IF X and Y are not independent then you need to worry about the correlation between X and Y (for more details about what is meant by the correlation between two random variables see Section 2.2).