Question

Let X denote the number of dad jokes Albert makes in one day. Suppose X has the following probability distribution: x f(x) 0 0.1 1 0.2 2 0.2 3 0.3 4 0.2 a) Find the probability that Albert will tell at least 2 dad jokes in a day. b) Find the expected number of times Albert will tell a dad joke in a day, E[X]. c) Find the standard deviation of dad jokes told in a day, sd[X].

Answer 1

a) 0.7

b) 2.3

c) 1.2689

Explanation:

a) Find the probability that Albert will tell at least 2 dad jokes in a day.

P(X ≥ 2) = P(X=2)+P(X=3)+P(X=4) = 0.2+0.3+0.2 = 0.7

b) Find the expected number of times Albert will tell a dad joke in a day, E[X].

E[X] = 0*P(X=0)+1*P(X=1)+2*P(X=2)+3*P(X=3)+4*P(X=4) =

= 0.2+2*0.2+3*0.3+4*0.2 = 2.3

c) Find the standard deviation of dad jokes told in a day.

The standard deviation can be defined as

`\sigma=√(E[X^2]-(E[X])^2)`

On one hand we have

`E[X^2]=0^2*0.1+1^2*0.2+2^2*0.2+3^2*0.3+4^2*0.2=6.9`

on the other hand,

`(E[X])^2=(2.3)^2=5.29`

So

`\sigma=√(6.9-5.29)=1.2689`