Statistics Continuous Random Variables

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Questions and Answers

What is the condition that a probability density function (PDF) must meet regarding its integral over the real line?

The integral of the PDF must be greater than 1.

The integral of the PDF must be less than 1.

The integral of the PDF must equal 0.

The integral of the PDF must equal 1. (correct)

What does the cumulative distribution function (CDF) represent for a continuous random variable X?

The probability density at a specific value of X.

The probability that X is less than or equal to a specific value. (correct)

The area under the PDF curve for all values.

The probability that X is equal to a specific value.

Which of the following statements about a continuous random variable X is correct?

P(X = a) is always greater than 0 for any real number a.

P(a ≤ X ≤ b) is the area under the PDF from a to b. (correct)

P(X < a) is equal to P(X ≤ a).

The mean of X is always equal to the median of X.

If the derivative of the cumulative distribution function (CDF) F(x) exists, what does it represent?

The probability density function (PDF) f(x). Signup and view all the answers

What is true about the variance of a continuous random variable X?

Variance measures the expected value of the squared deviations from the mean. Signup and view all the answers

What is the formula for the mean or expected value of a continuous random variable X?

$μ = E(X) = ∫ x ⋅ f(x) dx$ Signup and view all the answers

How is the variance of a continuous random variable X defined?

$V(X) = ∫x^2 f(x) dx - μ^2$ Signup and view all the answers

What expression represents the standard deviation of X in terms of variance?

$σ = √(V(X))$ Signup and view all the answers

Given the cumulative distribution function F(x), how do you find the probability density function f(x)?

$f(x) = F'(x)$ Signup and view all the answers

If Y = 3X + 1, how is the expected value E(Y) related to the expected value E(X)?

$E(Y) = 3E(X) + 1$ Signup and view all the answers

For the continuous random variable defined by $f(x) = 0.125x$ for $0 < x < 4$, how do you determine the mean?

Calculate $E(X) = ∫0^4 x(0.125x) dx$ Signup and view all the answers

What is the calculation for P(X < -2) given the piecewise cumulative distribution function?

0 Signup and view all the answers

If the variance of X is defined as $σ^2 = V(X)$, which option correctly describes its meaning?

It represents the average of the squared deviations from the mean. Signup and view all the answers

Study Notes

Continuous Random Variables

A continuous random variable can be any value within a given range
Its range can be finite or infinite
It's represented by an interval of real numbers

Probability Density Function (PDF)

A function that describes the probability distribution of a continuous random variable
Represents the probability of a random variable taking on a specific value within a given range
Has the following properties:
- The function is always greater than or equal to zero
- The area under the curve between negative infinity and positive infinity is equal to 1
- The probability of a random variable falling between two values is equal to the area under the curve between those two values

Cumulative Distribution Function (CDF)

Represents the probability that a continuous random variable takes a value less than or equal to x
Can be used to calculate the PDF for a continuous random variable if the derivative exists

Mean and Variance of a Continuous Random Variable

Mean (µ or E(X))
- The average value of the random variable
- Calculated by integrating x multiplied by the PDF
- Represents the expected value of the variable
Variance (σ² or V(X))
- Measures how spread out the distribution is
- Calculated using the formula: ∫(x - µ)² ⋅ f(x) dx or ∫x² ⋅ f(x) dx - µ²
Standard deviation (σ)
- The square root of variance
- Represents the average deviation of the variable from the mean

Expected Value

Can be calculated for a function of a continuous random variable
Formula is: ∫h(x) ⋅ f(x) dx where h(x) is the function and f(x) is the PDF
Represents the expected value of the function of the random variable

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Description

This quiz focuses on the concepts of continuous random variables, including their probability density functions (PDF) and cumulative distribution functions (CDF). Explore the properties of PDFs, the significance of areas under the curve, and the calculation of mean and variance. Test your understanding of these fundamental statistics topics.