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) = c * x * √(25 - x²) over [0, 5]

Accepted Answer

Hello there. Today we are going to solve the following practice from 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 constant k so that fx is equal to k multiplied by x multiplied by the square root of 16 minus x 2 is a probability density function on square brackets of 0.4. OK. So it appears for this particular prompt we're asked to determine what the value for the constant of k will be such that F of X will be specifically our function f x will be a probability density function on the closed interval because that's what the square brackets mean is square brackets indicate a closed interval, and we're asked to determine. How this constant, this value for constant K, such that F of X will be a probability density function on this closed interval of 0.4. So, to put it simply, using all the information that is provided to us, we're ultimately solving for the value of constant K. So, what does K represent? What is the value of K? So now that we know what we're ultimately trying to solve for. We need to recall and note that in order to solve for this problem, we will need to satisfy the two fundamental conditions for a valid probability density function, which a probability density function is denoted as P. DF. So we need to figure out how we will satisfy these two fundamental conditions for a valid probability density function. So considering non-negativity first, the function must be zero or positive for all values in its domain, meaning that f of x must be greater than or equal to 0. And for the normalization element, the total area under the curve over the given interval must equal exactly 1. For example, the integral evaluated from A to B of FXDX must be equal to 1. So, considering our problem specifically for this particular case, for the interval, which will be the closed interval 0.4, the term. X multiplied by the square root of 16 minus x 2. Is a naturally non-negative value. Because X starts at 0 and the square root of any real number is non-negative. And since the function is already positive, we can focus entirely on the second condition, meaning we will need to integrate the function from 0 to 4 and set the result equal to 1, and then at that point we will need to solve for the value of k in order to ensure the probability is indeed 1. So with that said, our first step is we need to set up our integral, and the definition of a probability density function requires that the integral evaluated from A to B of FX is equal to 1. So that will mean for this particular problem, we can write that the integral evaluated from 0 to 4 of K multiplied by x multiplied by the square root of 16 minus x to the power of 2. DX will be equal to 1. Fantastic. Our second step now is we need to solve for the, once again we're solving for the integral of k multiplied by x multiplied by the square root of 16 minus x to the 2 dx, and in order to solve for this integral, we will need to recall and use the u substitution method. So let us state that for this particular case, you will be equal to 16 minus x 2. And that will mean that DU by DX will be equal to -2X, and this is how we will get this answer by differentiating. So when we differentiate. We will find that DU by DX will be equal to -2X, which will mean that DU will be equal to -2XDX, which will mean that X multiplied by DX will be equal to 1/2 DU. Fantastic. And we now need to change our limits, so that will mean that when x is equal to 0, that will mean that U is equal to 16 minus 0 to 2, which is equal to 16, and all we're doing is we're plugging in a value of 0 in for x. So that will mean, similarly, when X is equal to 4, you will be equal to 16 minus 42, which will be equal to 0. And now we need to substitute our new variables into our integral, so we need to take the integral evaluated from 16 to 0 of K multiplied by the square root of U multiplied by parentheses of 1/2. DU which will be equal to 1. So to simplify our integral, we can flip the limits of integration from 0 to 16, and we can remove the negative sign, which will mean we can go ahead and write k divided by 2 multiply. By the integral of 0 to 16, which we can say that k divided by 2 multiplied by the integral evaluated from 0 to 16 of u to the power of 1/2 duu will be equal to 1. Our third step now is we need to recall and apply the power rule for integration. So we need to take k divided by 2 multiplied by square brackets of u to the power of 3 divided by 2, all divided by 3 divided by 2. Evaluated from 0 to 16 is equal to 1, fantastic. So we can write that k divided by 2 multiplied by 2 divided by 3 multiplied by square brackets of u to the power of 3 divided by 2, evaluated from 0 to 16 will be equal to 1. Which k divided by 3 multiplied by parentheses of 16 to 3 divided by 2 minus 03 divided by 2 will be equal to 1. And we need to note that since 16 to the power of 3 divided by 2 is equal to parentheses of the square root of 16 to the power of 3, which will be equal to 4 to 3, which is equal to 64, we can go ahead and write that k divided by 3 multiplied by 64 will be equal to 1. OK. So that will mean when we rearrange this expression to isolate and solve for K, which you can use algebra to do this, I'm gonna skip a few steps, but when you isolate and so solve for K by itself, you will find that K is simply equal to 3 divided by 64 when we write our final answer as a fraction. And that's it. That is our final answer. Hooray, we did it. So looking at the prom itself, we could safely say that our final answer for the variable constant k will be once again 3 divided by 64. And that's it. That is our final answer. 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) = c * x * √(25 - x²) over [0, 5]

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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) = c * x * √(25 - x²) over [0, 5]