Let X have the density function f(x) = 0.75(1

Let X have the density function $f(x) = 0.75(1 - x^2)$ if $-1 le x le 1$ and zero otherwise. 1. Find the distribution function. 2. Find the probabilities $P(-frac{1}{2} le X le frac{1}{2})$ and $P(frac{1}{4} le X le 2)$. 3. Find x such that $P(X le x) = 0.95$

Transcript

00:01 Hello everyone, we will go to solve problem based on density function statistics basically so we are given x -over density function f x equal to this is x f x equal to 0 .751 minus x squared like if x is less than greater than equal to 1 and less than 1 and it is 0 otherwise like this is given in the case now we have to find so our 34 equal to y this becomes integration of x to y and x y should lie between minus 1 and 1 this becomes fx d x now we have to find the probabilities p minus half to half this will be minus half to half half f x is 0 .75 0 .75 will basically comes out 1 minus x square b x this becomes 0 .75 minus half 2 half just basically you will do the integration 0 .75 of 1x x becomes x minus x cubed by three and this will be minus half to half solving this 0 .705 half minus minus xq 1 by 8 into 3 that becomes 24 minus minus minus half cube because minus 1 by 8 minus 1 by 24 so that will become plus solving this this becomes 12 by 24 minus 11 by 24 minus this will be minus 11 by 24 so this becomes 22 by 24 solving this this becomes this becomes 22 divided by 24 multiply by 0 .75 this becomes 0 .68 and 5 this becomes a final answer for this part now we have to find similar thing for minus 1 by 4 to 2 we will use the integration that we did so it will be 0 .75 x minus x squared by 3 it will be 1 by 4 to 2 and this will become 2 minus 8 by 3 minus 1 by 4 minus 1 by 64 into 3 192 this will be this becomes minus 2 by 3 minus it will be 4 40 multiply to 4 that it becomes 192 that is 50 minus 48 so so when you multiply by 48, 48 minus 1 is 47, 47 by 192.

06:16 Now what multiply to 3? that will be 64.

06:22 64, so minus 128 minus 47 divided by 92.

06:31 0 .705 is coming all the way.

06:35 So this will be 0 .75.

06:44 Multiplied by minus now use the calculator to calculate this 175 divided by 192 multiplied by 0 .75 this is 1 .6835 .16835 so this becomes the answer for this part now find x is that b x less than equal to x is 0 .95 now we have to find and b x less than equal to x is 0 .95.

07:26 0 .95.

07:29 Now this, so we did a mistake here.

07:47 1 by 4 to 2, this will not be 2 as maximum limit is 1.

07:52 So this will be, i will correct it...

Question

Let X have the density function $f(x) = 0.75(1 - x^2)$ if $-1 le x le 1$ and zero otherwise. 1. Find the distribution function. 2. Find the probabilities $P(-frac{1}{2} le X le frac{1}{2})$ and $P(frac{1}{4} le X le 2)$. 3. Find x such that $P(X le x) = 0.95$

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