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Answer:
Y will arrive earlier than X one fourth of times.
Step-by-step explanation:
To solve this, we might notice that given that both events are independent of each other, the joint probability density function is the product of X and Y's probability density functions. For an uniformly distributed density function, we have that:
Where L stands for the length of the interval over which the variable is distributed.
Now, as X is distributed over a 1 hour interval, and Y is distributed over a 0.5 hour interval, we have:
.
Now, the probability of an event is equal to the integral of the density probability function:
Where A is the in which the event happens, in this case, the region in which Y<X (Y arrives before X)
It's useful to draw a diagram here, I have attached one in which you can see the integration region.
You can see there a box, that represents all possible outcomes for Y and X. There's a diagonal coming from the box's upper right corner, that diagonal represents the cases in which both X and Y arrive at the same time, under that line we have that Y arrives before X, that is our integration region.
Let's set up the integration:
We have used here both the independence of the events and the uniformity of distributions, we take the 2 out because it's just a constant and now we just need to integrate. But the function we are integrating is just a 1! So we can take the integral as just the area of the integration region. From the diagram we can see that the region is a triangle of height 0.5 and base 0.5. thus the integral becomes:
That means that one in four times Y will arrive earlier than X. This result can also be seen clearly on the diagram, where we can see that the triangle is a fourth of the rectangle.
