## Question 2 - Probability distribution with an infinite number of possible values Grocery shoppers on Sunday can be split into two main groups: daily (picking up a few things they need for the next day or two) and weekly (picking up all items they need for the coming week). Let X be the number of items people have in their shopping cart at checkout. The possible values of x are 1, 2, 3, (no real upper limit, so we'll take that to be "infinity"). ... After studying historical data, you find that the following shape does a decent job at describing the relative frequencies of `x`. $$Shape = xe^{-x/2} + 0.01x^2e^{-x/8} \qquad x=0, 1, 2, 3, \ldots$$ Shape = xe +0.01x-x/8 2 x = 0, 1, 2, 3,... ^ To convert the shape equation into the equation for the probabilities, it's not possible to do the standard shape conversion trick (`p $$Shape = xe^{-x/2} + 0.01x^2e^{-x/8} \qquad x=0, 1, 2, 3, \ldots$$ C/2 Shape = xe +0.01x-x/8 x = : 0, 1, 2, 3, ... To convert the shape equation into the equation for the probabilities, it's not possible to do the standard shape conversion trick (`p a) Use Wolfram-Alpha to find the value of `???` in the equation above. your work in Wolfram (paste into Word after knitting). Sanity check: **Response: ** (put screenshot here) Provide a screenshot of it's a little above 14.15. b) Although the values of `x` technically go up to infinity, define ` to be the integer sequence 1 to 80. Then, *using your equation* for $P(X=x)$ (after you've found the value of `???` in (a); no p c) Provide a bar chart of the PMF. Label the x-axis "Items in Cart" and the y-axis "Probability". One hump represents the daily shoppers; the other represents the weekly shoppers. ```{r Q2c} d) Use your PMF vectors `x` and `p` to find the probability that a shopper has between 4 and 20 items in their cart. Sanity check: a little under 47%. ``{r Q2d}