Let X be a continuous random variable with density...

Let X be a continuous random variable with density function $f_X(x) = egin{cases} 3e^{-3x} & x > 0 \ 0 & ext{else} end{cases}$ a. Verify that $f$ is a density function. b. Calculate $P(-1 < X < 1)$. c. Calculate $P(X < 5)$. d. Calculate $P(2 < X < 4 mid X < 5)$. e. Find the cumulative distribution function of X. f. Find the mean of X. g. Compute $E[e^{2X}]$ h. Compute the median of X. i. Find the 90th percentile of the distribution of X. j. Find $Var(X)$

Transcript

00:01 The probability density function of the random variable x is basically given as 3e power negative 3x.

00:07 When x is positive and 0 otherwise.

00:12 First, let's verify whether it's a legitimate period.

00:15 So that means we need to integrate was 0 to infinity because x is positive only we have a piece.

00:21 Negative it is even 0.

00:23 3e power negative 3x d x.

00:25 It should be 1.

00:26 So let's see.

00:27 What is integration of e power negative 3x? it is e power negative 3x divided by negative 3.

00:32 Plug in the upper and lower limits.

00:35 When you keep upper limit, if our negative infinity 0, minus when you keep lower limit, it can be 3 or 0 divided by negative 3, which is 1.

00:43 Yes, it's the legitimate medium.

00:45 The second question is what is the probability that minus 1 is less than x less than 1? for that, we need to do integration from minus 1 to 1 the period.

00:54 3e power minus 3x d x but i can split this as minus 1 to 0.

01:04 Sorry, it's not 3, power minus 3x it's pdf minus 1 to 0 the pdf plus 0 to 1 the pdf but this is 0 because for negative part the pdf is 0 so this is 0 and for this we need to do the integration e power negative 3x integration 3 e power negative x integration is 3 power negative x divided by negative 3 basically it is negative e power negative 3x substitute the limits in 0 and 1 so you'll be getting the answer as 1 minus e power negative 3 this is that the answer to the second question.

01:43 And the probability that x is less than 5, obviously we should integrate from 0 to 5, 3 e power negative 3x d x, which is negative e par, negative 3x, substitute the limit to 0 and 5, so the answer is 1 minus e par negative 15.

01:59 So this is the answer for the third question.

02:05 The next question is the probability that 2 less than x less than 4, given that, the conditional property given that x is less than five.

02:16 So obviously by definition of conditional property, it will be probably that 2 less than x less than 4 comma x less than 5, that is the intersection divided by probably that x is less than 5.

02:29 Obviously the intersection of 2 less than x less than 4 and x less than 5 is 2 less than 4 divided by probability that x less than 5 is just from we calculated...

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