Random Variables and Distribution Functions 1

Problem 1.A Gumbel random variableXhas the distribution functionF_X(x) = exp(e^x).

Problem 1 Part a)The 0.5 quantile ofXoccurs whenx= 0.3665sinceF_X(0.365) = 0.5. Additionally, the 0.75 quantile ofXoccurs whenx= 1.2459since F_X(1.2459) = 0.75. Put another way, the median and the third quartile ofXare the valuesx= 0.3665andx= 1.2459, respectively.

Confirm by-hand the calculation of the first quartile,x=0.3266. Include all necessary steps in your calculation.

Problem 1 Part b)Using R, plot the graph ofF_X(x), indicating the first quartile, median, and third quartile as points on the plot. Label the axes appropriately. Below this plot, explain whyF_Xis a valid cumulative probability distribution function.

Problem 1 Part c)Consider the table of values forxandF_X(x)forxequal to integers from -2 to 5, which has been generated in R within the R chunk below.

x_val F_X

-2 0.000617979

-1 0.065988036

0 0.367879441

1 0.692200628

2 0.873423018

3 0.951431993

4 0.981851073

5 0.993284702

Using the table above, determine the probabilityP{1< X4}andP{4< X}.

Problem 1 Part d)Determine the probability density functionf_X(x)for the cumulative distribution functionF_X(x) = exp(e^x).

Problem 1 Part e)Sketch the distribution function together with a separate sketch of the density function (side-by-side), indicatingP{1< X4}on BOTH plots.