Fundamentals of Probability and Statistics for Engineers

3.2.3 PROBABILITY DENSITY FUNCTION FOR CONTINUOUS RANDOM VARIABLES For a continuous random variable ,its PDF, is a continuous fu ...

function can be interpreted as the mass density (mass per unit length). There are no masses attached to discrete points as in th ...

Let us compute some of the probabilities using The probability (0 1) is numerically equal to the area under from 0 to 1, as show ...

a case in which random variable X is continuously distributed over the real line except at X 0, where P(X 0) is a positi ve quan ...

For part (b), Figure 3.9 shows pX (x) for the discrete portion and fX (x) for the continuous portion of X. They are given by: an ...

and, for part (b), These results are, of course, the same as those obtained earlier using the PDF. 3.3 Two or More Random Variab ...

U sing again the mass distribution analogy, let one unit of mass be distributed over the (x,y) plane in such a way that the mass ...

first quadrant. When both X and Y are continuous, FX Y (x, y) becomes a smooth surface with the same features. It is a staircase ...

discrete. Let X and Y be two discrete random variables that assume at most a countably infinite number of value pairs (xi,yj),i, ...

For pX Y (4,1), there are five distinct ways of reaching that position (4 steps in the x direction and 1 in y; 3 in the x direct ...

and These are marginal pmfs of X and Y. The joint probability distribution function FX Y (x,y) can also be constructed, by using ...

Example 3.6.Let us discuss again Example 2.11 in the context of random variables. Let X be the random variable representing prec ...

density function (jpdf) of two ra ndom variables, X and Y, is defined by the partial derivative Since FX Y (x,y) is monotone non ...

We also note the following important properties: Equation (3.27) follows from Equation (3.25) by letting and this shows that the ...

The probability we are seeking is thus the volumeunder this surface over an appropriate region R. For this problem, the region R ...

Note that, for a more complicated jpdf, one needs to carry out the volume integral for volume calculations. As an exercise, let ...

Similarly, Both random variables are thus uniformly distributed over the interval (0, 60). Ex ample 3. 8. In structural reliabil ...

In closing this section, let us note that generalization to the case of many random variables is again straightforward. The join ...

which is expected. It gives the relationship between the joint jpmf and the conditional mass function. As we will see in Example ...

N ow we see that Equation (3.38) leads to which is in a form identical to that of Equation (3.35) for the mass functions – a sat ...