Two methods of measuring surface smoothness are used to...

Two methods of measuring surface smoothness are used to evaluate a paper product. The measurements are recorded as deviations from the nominal surface smoothness in coded units. The joint probability distribution of the two measurements is a uniform distribution over the region $0<x<4,0<y$, and $x-1<y<x+1 .$ That is, $f_{X Y}(x, y)=c$ for $x$ and $y$ in the region. Determine the value for $c$ such that $f_{X Y}(x, y)$ is a joint probability density function. Determine the following: (a) $P(X<0.5, Y<0.5)$ (b) $P(X<0.5)$ (c) $E(X)$ (d) $E(Y)$ (e) Marginal probability distribution of $X$ (f) Conditional probability distribution of $Y$ given $X=1$ $(\mathrm{g}) E(Y \mid X=1)$ (h) $P(Y<0.5 \mid X=1)$

Two methods of measuring surface smoothness are used to evaluate a paper product. The measurements are recorded as deviations from the nominal surface smoothness in coded units. The joint probability distribution of the two measurements is a uniform distribution over the region $0<x<4,0<y$, and $x-1<y<x+1 .$ That is, $f_{X Y}(x, y)=c$ for $x$ and $y$ in the region. Determine the value for $c$ such that $f_{X Y}(x, y)$ is a joint probability density function. Determine the following:
(a) $P(X<0.5, Y<0.5)$
(b) $P(X<0.5)$
(c) $E(X)$
(d) $E(Y)$
(e) Marginal probability distribution of $X$
(f) Conditional probability distribution of $Y$ given $X=1$
$(\mathrm{g}) E(Y \mid X=1)$
(h) $P(Y<0.5 \mid X=1)$

Key Concepts

Joint Probability Density Function

A joint probability density function (pdf) describes the likelihood of two continuous random variables occurring simultaneously. It provides a function over a two-dimensional space such that the probability of the variables falling within any region is given by the double integral of the pdf over that region. The function must be nonnegative and integrate to 1 over its entire domain.

Uniform Distribution

A uniform distribution over a region means that the probability density is constant across the specified region. This implies that every subset of the region with equal area (or volume in higher dimensions) receives an equal amount of probability. It simplifies many probability calculations as the density function takes on a single value within the defined limits.

Normalization Constant

The normalization constant (often denoted as c) is the constant value in a probability density function that ensures the total probability over the entire space is equal to 1. To determine this constant, one integrates the unnormalized pdf over the entire region of interest and then solves for the constant that makes the integral equal to 1.

Marginal Distribution

The marginal distribution of one variable is obtained by integrating the joint probability density function over the other variable. This process effectively 'sums out' the other variable, resulting in a probability density function that describes the behavior of a single variable irrespective of the other.

Conditional Distribution

The conditional probability distribution of one variable given another is derived from the joint pdf and the marginal pdf. It is defined as the ratio of the joint probability density to the marginal probability density of the given variable, and it describes the distribution of one variable when the value of the other variable is known.

Expected Value

The expected value, or mean, of a continuous random variable is a measure of its central location and is calculated by taking the integral of the variable multiplied by its probability density function over its entire range. It represents the long-run average outcome of repeated experiments.

Integration over Specified Regions

When dealing with joint probability distributions, it is often necessary to perform integration over a region defined by inequalities. This involves carefully setting up the limits of integration according to the boundaries of the region, which can be irregular, to correctly calculate probabilities and expected values associated with the random variables.

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