# 0+File C:/Users/jsole/Downloads/Homework3.pdfCzaanduma am ame Suvu1 iranin I emanaid ama aane n enAmcanh m aMem eHAranneuoutput and discussion are placed together (when R is needed). Make it clear what parts of theoutput are relevant and show how they answer the questions posed in the homework. You areencouraged to work together on the homework, but do not copy any part of a homework. Eachstudent must produce his/her own homework to be handed in1. Suppose X is a random variable taking values between 0 and 4, with density curve f (x)displayed belowC234-1 0 15(a) Determine the value of C in the plot.(b) Calculate P(X < 1).(c) Calculate P(1 < X < 3)(d) What is the median of X?2. The file samples.RData includes four samples of size 1000 from distributions with mean 1 and10:33 PM9/15/2019O Type here to search18

Question

Hello,

Please find attached the question I would like help on. Specifically part b.

I have already figured out the answer to part a is 1/2 and I think if I can figure out part b, I can do part c.

Step 1

a)

Total area under the curve is a triangle with area = 1/2 * base * height = 1/2 * 4 * C = 2C

Since you've figured out part a), you must already know that the total area under the density curve = 1

Therefore, 2C = 1 which gives us C=1/2

Step 2

b)

P(x<=1) means percent of area under the density curve which is less than or equal to 1 as shown by yellow triangle.

To calculate the area of the yellow traingle, we need its height which can be calculated by splitting the total areas into 3 parts.

Step 3

Area of region I = 1/2 * (1-0)* x  = 0.5x

Area of region 2 = (4-1)*x = 3x

Area of region 3 = 1/2 *(4-1)*(0.5-x) = 0.75-1.5x

Sum of all three areas must be equal to 1

Therefore,  0.5 x + 3 x + 0.75 -1.5x =1  or

&nbs...

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