Random Variables(KA/SAP/U9RandomVariables)

Unit 9: Random Variables

Topics:

• Discrete random variables
• Continuous random variables
• Transforming random variables
• Combining random variables
• Binomial random variables
• Binomial mean and standard deviation formulas
• Geometric random variables
• More on expected value
• Poisson distribution

Discrete Random Variables

A discrete random variable is a variable that can take on a countable number of possible values. Examples include the number of heads in 10 coin tosses, or the number of students present in a class. The probability distribution of a discrete random variable lists each possible value and its probability.

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Continuous Random Variables

A continuous random variable can take on any value within a given range. Examples include the height of students or the time it takes to run a race. The probability of a continuous random variable taking any exact value is zero; instead, we talk about the probability of it falling within an interval, described by a probability density function (PDF).

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Transforming Random Variables

Transforming a random variable means applying a function to it, such as scaling or shifting. If Y = aX + b, where X is a random variable, then Y is a transformed random variable. The mean and variance of Y can be found using the formulas:

• Mean: E[Y] = aE[X] + b
• Variance: Var(Y) = a²Var(X)

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Combining Random Variables

When combining two independent random variables X and Y, the mean and variance of their sum or difference can be found as follows:

• Mean: E[X ± Y] = E[X] ± E[Y]
• Variance: Var(X ± Y) = Var(X) + Var(Y) (if X and Y are independent)

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Binomial Random Variables

A binomial random variable counts the number of successes in a fixed number of independent trials, each with the same probability of success. The probability of getting exactly k successes in n trials is given by:

P(X = k) = C(n, k) p^k (1-p)^n-k

where C(n, k) is the binomial coefficient.

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Binomial Mean and Standard Deviation Formulas

For a binomial random variable X ~ Binomial(n, p):

• Mean: μ = np
• Standard deviation: σ = √(np(1-p))

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Geometric Random Variables

A geometric random variable counts the number of trials needed to get the first success in a sequence of independent trials, each with the same probability of success p. The probability that the first success occurs on the k-th trial is:

P(X = k) = (1-p)^k-1 p

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More on Expected Value

The expected value (mean) of a random variable is the long-run average value of repetitions of the experiment it represents. For a discrete random variable X with possible values x_i and probabilities p_i:

E[X] = Σ x_ip_i

For a continuous random variable, the expected value is the area under the curve of x times the PDF.

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Poisson Distribution

The Poisson distribution models the number of times an event occurs in a fixed interval of time or space, when these events happen with a known constant mean rate and independently of the time since the last event. The probability of observing k events is:

P(X = k) = (λ^k e^-λ) / k!

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