Write a program in R which does the following: Generates the first 50 terms of the sequence beginning 0, 0.05, 0.1, 0.15 (ie which increases by 0.05 at each step): [math: x1,x2,…,x50] Derives the values of the pdf for the exponential distribution with mean 2, for each element of the sequence in step 1: [math: y1,y2,…,y50] Calculates [math: z=∑i=150xiyi] Plot [math: x1,x2,…,x50] against [math: y1,y2,…,y50], displaying the 50 points together with a line that joins them up and showing appropriate labelling. Do not worry about setting a working directory. When the time runs out, you will have an extra 3 minutes to upload your R file using the link below.

Question

BlackTom AI · Accepted Answer

The question presents an essay-style prompt asking for an R program that performs four tasks: (1) generate the first 50 terms of a sequence starting at 0 and increasing by 0.05, (2) derive the PDF values for an exponential distribution with mean 2 for each sequence element, (3) compute z as the sum of xi * yi for i = 1 to 50, and (4) plot the x values against the corresponding y values with a connected line and proper labels. The provided option contains a complete R script that addresses all these steps, so we can evaluate its contents against each requirement.

Option content analysis:
- Step 1: Generating the sequence
  The script uses x <- seq(0, by = 0.05, length.out = 50). This correctly creates the first 50 terms beginning at 0 with increments of 0.05, yielding x1, x2, ..., x50 as required. The seq function with by = 0.05 and length.out = 50 is an idiomatic and reliable way to construct this sequence.
- Step 2: Calculating the PDF for the exponential distribution with mean 2
  The exponential distribution with mean 2 has rate lambda = 1/mean = 0.5. The code sets lambda <- 1 / 2 and then computes y <- dexp(x, rate = lambda). This correctly evaluates the PDF at each xi for the specified exponential distribution, producing y1, y2, ..., y50 as required. The dexp function appropriately handles the vector input x, returning a vector of PDF values.
- Step 3: Computing z = sum of xi * yi
  The line z <- sum(x * y) performs the elementwise multiplication of the corresponding x and y values and then sums them, which matches the mathematical definition z = sum_{i=1}^{50} xi yi. This is implemented succinctly and efficiently.
- Step 4: Plotting x against y with a connected line and labels
  The script uses plot(x, y, type = 'b', pch = 19, col = 'blue', xlab = 'x (sequence values)', ylab = 'y (Exponential PDF at x)', main = 'Exponential PDF Values by Sequence'). grid() adds a grid for readability. The type = 'b' option displays both points and lines, which provides the requested connected visualization of the 50 points. The axis labels and title are present, satisfying the labeling requirement. While the grid() call is optional, it enhances readability and does not conflict with the task.

Additional notes:
- The comment line explicitly clarifies the interpretation of the rate parameter as 0.5, reinforcing the link between mean and rate for an exponential distribution.
- The final line in the answer block remarks that the script fulfills all requirements, which aligns with the four tasks listed in the prompt.
- There is no dependence on setting a working directory, as requested in the prompt, so the script is portable in that sense.

Potential edge considerations (not required but worth mentioning):
- If the user desires different labeling or a different plot style, the script can be easily extended, for example by adding a legend, or saving the plot to a file with png() or pdf().
- The code prints the value of z to the console via cat, which is helpful for immediate feedback but could be omitted or redirected if not desired.

In summary, for each required step, the option provides correct and efficient R constructs: generating the sequence with seq, computing the exponential PDF with dexp and the appropriate rate, aggregating the weighted sum with sum(x * y), and plotting with a clear, labeled, connected scatter-plot. The approach is coherent, concise, and faithful to the problem specifications.

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