# Exercise 3: Probability Distributions Planetary scientists might be interested in knowing the number of planets associated with egtrgsglgr planetary systems (planetary systems associated with stars other than our sun, Sol). Imagine that a large, well-funded research project has assembled multiple planetary research teams and charged each team with studying a separate set of ten known planetary systems (stars with at least one known planet), randomly assigned to each group. After intensive study, each team reports the number ($k$) of planetary systems they have detected that have at least two planets. Here, $k$ is a sample statistic: a count summarizing the number of observations out of 10 that have two or more planets: $$ k_j=\\sum_{i=1}A{10}x_{i,j}$$ >> where $j$ is a placeholder for the index or name of a particular research team, $i$ is a placeholder for the index of one of the planetary systems studied by that team, and $x_{i,j}$ is an indicator variable equaling 1 if the $i$th planetary system studied by the $j$th research team has two or more identifiable planets. Assuming that 22% of detectable planetary systems have two or more planets detectable from earth using current technology, the table below represents a model probability distribution for the statistic $k$. `{r, echo = FALSE} kable( matrix( data = c( 0:10 round(100*c(8. 335776e-02, 2. 351116e-01, 2.984109e-01, 2. 244458e-01, 1. 107842e-01, 3. 749617e-02, 8. 813203e-03, 1. 420443e-03, 1.502392e-04, 9. 416700e-06, 2. 655992e-07), 2) nrow = 11, ncol = 2, byrow = FALSE, dimnames = list( 0:10, c("k", "percentage of teams") row. names = FALSE k percentage of teams 8.34 23.51 29.84 22.44 11.08 0 6 0 0 V a U A W N - O 3.75 0.88 0.14 0.02 0.00 0.00(A). Roughly, what is the shape of this model probability distribution? (Recall the features of distributional shape we looked at during our discussion of sample distributions, including asymmetry or skewness, direction of skew, and number of modes.) *Write the shape in a complete sentence*