The random variable x is known to be uniformly distributed between 1.0 and 1.5 . a. Show the gra

The random variable x is known to be uniformly distributed...

Key Concepts

Continuous Uniform Distribution

The continuous uniform distribution is a probability distribution where all outcomes in the interval are equally likely. It is defined over an interval [a, b], and every subinterval of equal length within [a, b] has the same probability. This concept is fundamental for understanding situations where there is no weighting towards any specific outcome within a bounded range.

Probability Density Function

The probability density function (PDF) for a continuous random variable describes the relative likelihood of the variable taking on a particular value. For a uniform distribution on [a, b], the PDF is a constant function equal to 1/(b - a) over the interval and zero elsewhere. Graphing this function results in a horizontal line over the interval, representing that the probability is uniformly distributed.

Point Probability in Continuous Distributions

In the context of continuous probability distributions, the probability of the random variable taking on any specific single value is zero. This is because the probability is defined as an area under the curve of the probability density function, and a single point has no area.

Interval Probability for Continuous Distributions

The probability that a continuous random variable falls within a certain interval is calculated by integrating the probability density function over that interval. For a uniform distribution, this simplifies to multiplying the constant PDF value by the length of the interval, giving a straightforward method to obtain the probability over any interval within the distribution's support.

Transcript

00:01 So we're given that some variable x is uniformly distributed between 1 and 1 .5.

00:10 So for part a, it wants us to graph or sketch a graph of this probability distribution.

00:17 So it's going to look something like this.

00:22 We know from our formula that's going to be 1 minus 1 .5 over, sorry, 1 over 1 .5 minus 1 for x being between 1 and 1 .5 and 0 elsewhere.

00:48 So if here's our x and f of x, and this is, if we actually do this right now, 1 minus 1 .5, sorry, 1 .5 minus 1 is 1 half, 1 over 1 half, is 2.

01:11 So if here's 1 .0, here's 1 .5, and then 2 will be up here.

01:21 And so the distribution will look like this.

01:34 It'll just be that area.

01:36 All right.

01:39 For the first thing, or sorry, for the first, for part b, we want to find the probability that we choose the point 1 .25 exactly.

01:55 And that's zero.

01:56 Because if you really think about it, 1 .25 is right here.

02:04 And this sliver here has a horizontal range of zero, essentially.

02:12 It's an infinitely thin sliver that we're looking at.

02:16 So the probability that we choose that exact point, when there's infinite other points along this line, is so small that we might as well call it zero.

02:26 It's infinitely small.

02:29 Now, we need to find the probability that x falls between 1 and 1 .25.

02:39 And that's just going to be, well, we know that this entire thing is 2 here, and this makes up half of this line down here...

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