How Can I Solve a Density Function Using the Fundamental Theorem of Calculus?

[barneygumble742]

hi,

i'm given the density function
fx(t) = 0 (t<0)
fx(t) = 2t (0<t<1)
fx(t) = 0 (t>1)

how can i solve this using the fundamental theorem of calculus?

i had a similar situation before where my function was:
fx(t) = 0 (t<0)
fx(t) = 1 (0<t<1)
fx(t) = 0 (t>1)

and the g(t) i came to was:
g(t) = 0
g(t) = t
g(t) = 2-t
g(t) = 0

some work from the previous situation:
integral of fx(u) * f(t-u) du
integral of 0 for t<0 = 0
integral of 1 for 0<t<1 = t
integral of 1 for 0<t-t<1 = 2-t for 1<t<2
integral of 0 for t>2 = 0

finding the probability between alpha and beta:

integral of g(t) dt from 0.45 to 1.35 = integal of t dt from 0.45 to 1 + integral of 1 to 1.35 = 0.6875
which is the correct answer.

i tried to apply the same principles where fx(t) = 2t but i keep getting the wrong answer.
for my simulation, x1 = rnd (a random number between 0 and 1) and x2 = sqrt(rnd) (square root of another random number). i added the numbers and I'm finding the expected value and variance perfectly and my theory supports it. however I'm not getting the theory for my probability of being between alpha and beta.

i think I'm loosin' it!

 

[lippyka]

any help would be greatly appreciated.using the fundamental theorem of calculus, you can solve this density function by taking the definite integral of fx(t) from 0 to t. The result will give you the cumulative density function (CDF) for fx(t). The CDF is defined as the probability of a random variable X being less than or equal to some value x. In this case, it would be the probability of a random variable T being less than or equal to a given value t. The CDF gives you the area under the density function curve up to the given value t. To find the probability of a random variable T being between alpha and beta, you would take the definite integral of fx(t) from alpha to beta. This integral gives you the area under the density function curve between alpha and beta, which is the probability of T being between those two values. For your particular density function, the CDF is given by: Fx(t) = 0 (t<0)Fx(t) = t^2 (0<t<1)Fx(t) = 1 (t>1)To find the probability of T being between alpha and beta, you would take the definite integral of fx(t) from alpha to beta. This would give you the area under the density function curve between alpha and beta. For example, if alpha = 0.45 and beta = 1.35, then the probability of T being between those two values would be: P(T ∈ [0.45,1.35]) = integral of fx(t) from 0.45 to 1.35 = integral of 2t from 0.45 to 1 + integral of 0 from 1 to 1.35 = integral of 2t from 0.45 to 1 = 0.6875