September 15, 2013

To begin, load some data into your **R** session by executing the following.

```
source("http://www3.nd.edu/~steve/computing_with_data/homework/hmwrk1Data.R")
```

Warm-up: Vectors of one class can sometimes be coerced into a vector of another class. Look up the help entries for `as.character`

, `as.numeric`

, `as.integer`

. Create vectors of varying types and experiment with the result of trying to coerce it to another type. (Donâ€™t forget logical vectors.)

Given \( x1 \) form a new vector \( y1 \) whose entries are the entries of \( x1 \) with odd index; i.e., \( x1[1] \), \( x1[3] \), ….

Form a vector \( y2 \) such that for each \( j \), \( y2[j]=x1[j] \), if \( x1[j] > 0 \), and \( y2[j] = 0 \), if \( x1[j] \leq 0 \).

Form a vector y3 such that for each \( j \), \( y3[j]=\log(x2[j]) \). Try the same with \( x1 \) and report the result. Note: This is the natural logarithm. log2 is the log base \( 2 \).

Given a discrete random variable \( X \), with values \( X_i \), for \( i \leq n \), let \( p_i \) denote the probability of \( X_i \). Let \( \mu \) denote the mean of \( X \); i.e., \( \mu=\sum_{i\leq n} X_ip_i \).The

*r-th moment about the mean of \( X \)*is \( \mu_r = \sum_{i\leq n}(X_i - \mu)^rp_i \). Consider \( x1 \) as a random variable with distribution \( p1 \). Compute \( \mu_3 \) of \( x1 \). (\( \mu_3 \) is related to the skewness of the distribution.)Create a vector \( y4 \) whose ith entry is \( 0 \) if \( x1[i]> 0 \) and \( 1 \) if \( x1[i] \leq 0 \).