statistics help

Question

The number of fatal car crashes per day in Houston is shown in the table below. Let X be the random variable representing the number of fatal crashes. Find P(X < 2). Hint: convert the table to a probability distribution first by dividing the "Days" column by the total numberof days.

Number of Fatal Crashes

0

1

2

3

4

5

Number of Days
144
172
128
37
7
1

Answer 1

I bet Maurizio;s answer is the one your teacher wants but is is surely not a very good answer.  In fact, it's one of those really disappointing questions because it misses an opportunity to talk about something beautiful.  The number of fatal car crashes in a day is a Poisson r.v..  It has to be if you believe only some very small assumptions about how fatal car crashes happen.  That means you estimate the Poisson mean with the data above (0*144 + 1*172+...)/489 and then calculate Poisson probabilities using the formula you can get from Wiki.  That would give about 0.673 compared to Maurizio's answer of 0.645 which isn't much different here.

 

The Poisson estimate is much more efficient (i.e., uses the data better).  Among other things Maurizio's estimate treats a day with 4 crashes in the same way it treats a day with 5 crashes so important information is lost. 

 

It's awful to work hard at convincing people that statistics is boring...

Answer 2

You are provided with the right hint.

Let X = number of fatal crashes.  You can derive the probability associated with X by using the Number of days

Total number of days = 144+172+128+37+7+1 = 489 (check the calculations!)

 

Prob. [X=0] = 144/489 = 0.2945

Prob. [X=1] = 172/489 = 0.3517

 

and so on..

 

To find the Prob[X<2], you need to recognize that this is the same as Prob.[X=0] + Prob.[X=1] and ... you are done.