Random Variables and Probability Distribution

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 +∞.