FMAT prob&stat continuous random variables incompl Flashcards
What is a continuous random variable?
A continuous random variable is a variable that can take any real value within a given range. Unlike discrete random variables, which take specific values, a continuous random variable can assume infinitely many values within an interval. Examples include the weight of a newborn baby, the lifetime of a light bulb, or the time taken to swim a length of a swimming pool.
How is the probability of a continuous random variable determined?
The probability of a continuous random variable falling within a certain range is determined using a probability density function (p.d.f.), denoted as f(x). The probability of the variable taking a specific value is always zero, but the probability of it falling within an interval is found by integrating f(x) over that interval.
What is the key property of a probability density function (p.d.f.)?
The total area under the probability density function must be equal to 1. This ensures that the sum of all possible probabilities for the variable equals 1, representing certainty that some outcome will occur.
How do you calculate the probability that a continuous random variable falls within a given interval?
The probability that a continuous random variable X falls within the interval [a, b] is given by the integral:
P(a ≤ X ≤ b) = ∫[a to b] f(x) dx.
This integral computes the area under the curve of f(x) from x = a to x = b.
What is a piecewise-defined probability density function?
A piecewise-defined probability density function is a function that has different expressions for different ranges of x. This means that f(x) is defined by separate formulas depending on which part of the domain x falls into.
How do you determine unknown constants in a piecewise-defined p.d.f.?
To determine unknown constants in a piecewise-defined function, we use the condition that the total probability must be 1. This means integrating f(x) over all possible values of x and setting the result equal to 1:
∫ f(x) dx = 1.
If the function has different expressions in different intervals, the integral is computed separately for each interval, and their sum must equal 1.
What is the expectation (expected value) of a continuous random variable?
The expectation (or mean) of a continuous random variable X, denoted as E(X), is the weighted average of all possible values of X. It is given by:
E(X) = ∫ x * f(x) dx.
This integral finds the balance point of the distribution, representing the long-run average value of X over many repetitions.
What is the expectation of a function of a continuous random variable?
If Y is a function of X, written as Y = g(X), then the expectation of Y is found by integrating g(X) multiplied by the probability density function of X:
E(Y) = ∫ g(x) * f(x) dx.
This generalizes expectation beyond just X itself, allowing us to compute expected values for transformed variables.
What is the expectation of a linear function of a random variable?
If Y is a linear function of X, given by Y = aX + b, then its expectation follows the rule:
E(aX + b) = aE(X) + b.
This means scaling a random variable by a factor a scales its expectation by a, and adding a constant b shifts the expectation by b.
Example: If a random variable X has an expectation of 5, what is the expectation of Y = 2X + 3?
Using the linearity property of expectation:
E(Y) = 2E(X) + 3 = 2(5) + 3 = 10 + 3 = 13.
How do you calculate the probability that a function of a random variable falls below a given value?
To find P(g(X) ≤ c) for some function g(X), integrate the probability density function f(x) over the values of x that satisfy g(x) ≤ c.
What is the meaning of a probability density function defined in terms of a polynomial or cubic function?
When a probability density function is expressed as a polynomial or cubic function, it means the probability distribution follows a curved shape rather than a simple linear or uniform distribution. The area under this function still must sum to 1, and probability calculations require integrating polynomial expressions.
How can the probability of an event be found using an alternative approach?
Instead of directly computing P(A), we can use the complement rule:
P(A) = 1 - P(A’).
This is useful when P(A’) (the probability of the event not happening) is easier to compute than P(A).
Why does expectation behave linearly for linear functions of a random variable?
Expectation follows linearity because integration (which defines expectation) is a linear operation. This means that scaling a variable scales its expected value proportionally, and adding a constant shifts the expected value without affecting its variability.
