Question

Suppose the random variable \(X\) has the cdf $$ F(x)=\left\\{\begin{array}{ll} 0 & x<-1 \\ \frac{x+2}{4} & -1 \leq x<1 \\ 1 & 1 \leq x \end{array}\right. $$ Write an \(\mathrm{R}\) function to sketch the graph of \(F(x)\). Use your graph to obtain the probabilities: (a) \(P\left(-\frac{1}{2}

Short answer

(a) 0.375 (b) 0 (c) 0 (d) 0

Step-by-step solution

Code the function F(x) in R

Use the 'ifelse' function in R to write an equivalent code for \(F(x)\).\n \nF <- function(x) {\n ifelse(x < -1, 0, ifelse(x < 1, (x+2)/4, 1))\n}\n\nHere, we define a function \(F\) in R which mimics the piecewise function given. 'ifelse' is a great function in R that helps to write such piecewise defined functions.

Plot the graph of F(x)

Generate values between -2 and 2 and get the CDF values. Then use these values to make the plot.\n\nx <- seq(-2, 2, 0.01)\ny <- sapply(x, F)\nplot(x, y, type='l')\n\nIn this code, the 'seq' function generates values between -2 to 2 with increments of 0.01. The 'sapply' function applies the function \(F\) to all values of \(x\). 'plot' is used to plot \(y\) against \(x\) where \(y\) represents the values of \(F(x)\). The option type='l' ensures the result is a line plot.

Compute the required probabilities

Use the graph to get the specified probabilities. \n\n(a) \(P\left(-0.5 < X \le 0.5\right) = F(0.5) - F(-0.5)\) \n(b) \(P(X = 0) = F(0) - F(0)\) \n(c) \(P(X = 1) = F(1) - F(1)\) \n(d) \(P(2<x \le 3) = F(3) - F(2)\)\n\nHere, the cumulative distribution function (CDF) properties are used. The probability that \(X\) lies in an interval \((a, b]\) is given by \(F(b) - F(a)\) where \(F\) is the CDF of \(X\). The probability that \(X\) takes a specific value is always 0 for a continuous random variable.