In an experiment with 10,000 electrons, which land...

In an experiment with 10,000 electrons, which land symmetrically on both sides of $x=0,5000$ are detected in the range $-1.0 \mathrm{cm} \leq x \leq+1.0 \mathrm{cm}, 7500$ are detected in the range $-2.0 \mathrm{cm} \leq x \leq+2.0 \mathrm{cm},$ and all 10,000 are detected in the range $-3.0 \mathrm{cm} \leq x \leq+3.0 \mathrm{cm} .$ Draw a graph of a probability density that is consistent with these data. (There may be more than one acceptable answer.)

Key Concepts

Probability Density Function (PDF)

A probability density function is a function that describes how the probability of a continuous random variable is distributed over its possible values. In this context, the PDF provides a way to visualize the likelihood of detecting electrons at different x positions, with the probability of detection in any given interval found by integrating the function over that interval.

Normalization Condition

Normalization ensures that the total probability across the entire domain is equal to 1 (or 100% of the events in the case of an experimental count). When constructing a probability density from experimental counts, such as electrons landing within certain ranges, it is necessary to adjust the area under the PDF so that it accurately represents the full set of outcomes.

Integration Over Intervals

The probability of detecting an electron within a specified range is obtained by integrating the PDF over that range. This concept is essential in linking the theoretical probability distribution to experimental data, as the integral over ranges like [-1, +1], [-2, +2], and [-3, +3] must match the observed electron counts.

Symmetry in Distributions

Symmetry in a probability distribution implies that the function is mirror-imaged about a central point, in this case x = 0. Such symmetry indicates that the likelihood of an electron being detected at a positive displacement is the same as at the corresponding negative displacement, which simplifies analysis and reflects the underlying uniform conditions of the experimental setup.

Transcript

00:01 So we are given an experiment with 10 ,000 electrons which land symmetrically on the positive side of the axis and the negative side of the x -axis.

00:12 So they want us to...

00:19 Let me do this better.

00:26 They want us to write a probability density to drop probability density.

00:31 We call it row of x.

00:35 So what do they tell us? they tell us that 5 ,000 electrons are detected between minus 1 and 1 centimeters.

00:46 So it would be between here and here.

00:53 And they don't tell us anything else.

00:55 So we could just draw a constant probability, density between minus 1 and 1.

01:05 But we also know that, well, 5 ,000, that's 1 half of the total number of electrons.

01:14 And we know that the area beneath the probability density from minus 1 to 1 needs to be equal to the probability of the electron landing between minus 1 up to 1.

01:37 So since we're drawing a constant probability density this will be a rectangle and the area of this rectangle needs to be 1 half since the base is 2.

01:49 We have that the height times 2 is 1 1⁄2 so the base is 1 4th sorry the height is 1 4th so 1 4th could be here just try it like this so this is the first branch the second branch um they're actually actually true branches right so it's between minus 2 and 2 we have in total 75 % of the electrons so since 50 % land between minus 1 and 1, then we have 25 % that land between minus 1, minus 2 and 1 and 2.

02:36 So, and since they land symmetrically in the x -axis, so the area beneath this part needs to be equal to the area beneath this part, and it's they add up 1 -4th, so it's 1 -8th.

03:02 Is equal to, well, the second height that you need times one...

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