For the distribution of Exercise 3, find (a) E(X) (b) E(X^2) (c) E(4 X+12) (d) var(X) (e) var(4

For the distribution of Exercise 3, find (a) E(X) (b) E(X^2)...

Key Concepts

Expected Value

The expected value of a random variable is a measure of its central tendency, representing the long-run average outcome of repeated experiments. It is calculated as the sum or integral of the possible values of the random variable weighted by their probabilities.

Second Moment

The second moment E(X²) of a random variable is the expected value of the square of the variable. This moment is crucial because it is used in calculating the variance and provides insight into the spread of the distribution beyond the mean.

Linearity of Expectation

Linearity of expectation is a key property which states that the expected value of a linear combination of random variables is equal to the same linear combination of their expected values. This property holds regardless of whether the random variables are independent.

Variance

Variance measures the dispersion of a random variable around its mean, quantifying how much the values of the random variable deviate from the expected value. It is defined as the average of the squared deviations from the mean, i.e., Var(X) = E(X²) - [E(X)]².

Variance under Linear Transformations

When a random variable undergoes a linear transformation of the form aX + b, its variance is scaled by the square of the coefficient a. This means that Var(aX + b) = a² Var(X), illustrating how scaling a random variable affects the spread of its distribution while a constant shift does not.

Transcript

00:01 We have here a continuous random variable x whose probability density function is given by this expression.

00:08 The first part of the problem which asks us to find the expectation value of x is straightforward.

00:14 We simply evaluate this integral for x going from 0 to 1, which is the support of the probability density function.

00:24 It's straightforward to evaluate and gives us 2 over 3 as the expectation.

00:30 Of x.

00:32 For the rest of the problem, we are going to consider a random variable y, which is given by x squared.

00:39 We are going to first find the probability density function for y, and then we are going to find the expectation of y in two different ways.

00:48 In this case, y as a function of x is differentiable and strictly increasing for x between 0 and 1.

00:58 So we can apply the general formula for finding the transformed distributions.

01:06 By finding the inverse function of x as a function of y, which is the square root of y.

01:16 Plugging it into the distribution function of x, we get 2 times square root of y.

01:27 We also need to compute the derivative of the inverse function, which in this case gives us combining these two parts, we get the probability density function for the transformed variable y is simply 1, and that is for y between 0 and 1.

01:58 Otherwise, the density function is just 0.

02:03 Now that we have the pdf of y, we can find its expectation using the standard...

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