Find the value of the constant c so that the given function is...

Question

Find the value of the constant c so that the given function is a probability density function for a random variable X over the specified interval.

f(x) = (1/x) over [c, c + 1]

Accepted Answer

Hello there. Today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Determine the value of p so that fx is equal to e to the power of x is a probability density function on the interval square brackets of P, P + 2. OK, so it appears for this particular problem, we are asked to determine what the value of P is such that F of X is equal to e to the power of x will be a probability density function on the interval, specifically the closed interval of P, P + 2, if square brackets indicate a closed interval. So now that we know that we're ultimately solving for this value of P. Let us note that based on our prior knowledge, we need to note that in order to be a valid probability density function that will mean that F of X must satisfy two conditions. Firstly, it must be a non-negative value, meaning that fx must be greater than or equal to 0. For all x within the interval and the total area under the curve or the definite. Must be equal to 1. So once again, so in order to be a valid probability density function, f of x must satisfy these two conditions. Firstly, it must be non-negative for all x within the interval, and the second condition, the total area under the curve or the definite, must equal 1. So, turning our attention to our problem itself specifically, we are told that F of X or function FX is equal to e to the power of X, and this is an exponential function. That is strictly positive for all real numbers, so that means not the non-negativity condition is automatically satisfied. Therefore, we can safely say that we only need to find the value of P that will satisfy the normalization condition by setting the definite integral over the closed interval of P, P + 221. So with that in mind, since we now know that we're setting our closed interval simply to 1, our first step that we need to take is we need to set up our integral condition, so we need to have the integral evaluated from P2 P + 2 of E to the power of XDX is equal to 1, fantastic. Our second step now is we need to integrate and apply our limits, so we need to take square brackets of e to the power of x, evaluated from P to P + 2 is equal to 1, which will give us e to the power of parentheses of P + 2 minus e to the power of P. is equal to 1, which will mean that e to the P minus e p + 2 will be equal to 1. Our third step now is we need to solve for P now. So in order to solve for P, we need to factor out our variable e to the power of P. So we can go ahead and write eP multiplied by parentheses of 1 minus e 2 is equal to 1. Which will mean that eP will be equal to 1 divided by 1 minus e to the -2. And we to get rid of our e variable, we need to take the natural log of both sides, and when we do, we will get P is equal to the natural log of 1 divided by 1 minus e to the power of -2. And we need to get rid of the negative on our P, and when we do, we'll get P is equal to negative, the natural log of 1 divided by 1 minus e to the power of -2. Which will give us a final answer of which is equal to the natural log of 1 minus e to the power of -2, and that's it. That is our final answer. Hooray, we did it. So looking at the prom itself to recap what we've learned, we could safely say that the value of p once again without a doubt is the natural log of 1 minus e to the power of -2, and that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Find the value of the constant c so that the given function is a probability density function for a random variable X over the specified interval.f(x) = (1/x) over [c, c + 1]

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Find the value of the constant c so that the given function is a probability density function for a random variable X over the specified interval.
f(x) = (1/x) over [c, c + 1]