4.07 Quiz: Continuous Random Variables & Discrete Probability Distribution Flashcards

Study Notes

Random Variables and Probability Distribution

A sample space is the set of all possible outcomes of an experiment.
A random variable is a function that associates a real number to each element in the sample space and can be determined by chance.

Discrete Random Variables

Discrete random variables can take on a finite number of distinct values.
Examples include the number of heads acquired while flipping a coin three times, the number of kin an individual has, and the number of students present in a study hall at a given time.

Continuous Random Variables

Continuous random variables can take an infinitely uncountable number of potential values, usually measurable amounts.
Examples include the height or weight of an individual, the time an individual takes to wash, time, temperature, item thickness, length, and so forth.

Steps in Finding the Value of a Random Variable

Determine the sample space.
Count the number of random variables in each outcome in the sample space and assign this number to this outcome.

Probability Distribution of a Discrete Random Variable

A discrete probability distribution or probability mass function consists of the values a random variable can assume and the corresponding probabilities of the values.
Steps to get the probability distribution of a discrete random variable:
- Determine the sample space.
- Determine the possible values of the random variable in the given sample space.
- Assign probability values to each of the possible values of the random variable.
- Construct a histogram for the probability distribution.

Mean and Variance of Discrete Random Variable

Mean of a discrete random variable (μ) refers to the central value/average of its corresponding probability distribution.
Formula for the mean: μ = ∑[X1P(X1) + X2P(X2) + … + XnP(Xn)]
Variance and standard deviation of a random variable describe how scattered or spread out the scores are from the mean value of the random variable.
Formula for the variance: σ² = ∑[(X - μ)²P(X)]
Formula for the standard deviation: σ = √σ²

Alternative Procedure in Finding the Variance and Standard Deviation

Find the mean of the probability distribution.
Multiply the square of the value of the random variable X by its corresponding probability.
Get the sum of the results obtained in step 2.
Subtract the mean from the results obtained in step 3.

Normal Distribution

The graph of a normal probability distribution is called the normal curve, characterized by its mean μ and standard deviation σ.

Properties of the Normal Curve

The normal curve is bell-shaped.
The total area under the normal curve is 1.
The curve is symmetrical about its center.
The mean, median, and mode are equal and coincide at the center.
The tails of the curve are plotted in both directions and flatten out indefinitely along the horizontal axis (asymptotic to the baseline).
The mean determines the location of the center while the standard deviation determines the shape of the graph (in particular, the height and width of the curve).

Empirical Rule

The probability that a normal random variable X equals a particular value a is zero, i.e., P(X = a) = 0.
The probability that X is less than a, P(X ≤ a), equals the area under the normal curve bounded by a and -∞.
The probability that X is greater than some value a, P(X > a), equals the area under the normal curve bounded by a and +∞.

Description

Learn about sample space, random variables, and discrete random variables in probability theory. Understand how to associate real numbers with outcomes and determine values by chance. Explore examples like flipping coins and counting students to grasp the concept.