Question

Question 1:
Given the following probability distribution for a random variable X: x P(X=x)
-2
0.30
-1
0.15
0
0.20
1
0.20
2
0.15
a) Explain two reasons why the above distribution is a valid probability distribution.
b) Calculate μX and σX.
c) Determine the cdf(X), and write it as an additional column in the table.
d) Calculate P(−1<X≤3) .
e) Draw a histogram that represents the probability distribution of X.

Answer 1

Question 1 :

The probability distribution of X is

x	-2	-1	0	1	2	Total
P(X=x)	0.30	0.15	0.20	0.15	0.15	1

a) i) Since for every value of X

ii) Total of all probabilities is 1.

Hence given probability distribution is valid.

b)

x	p	x*p	x2*p
-2	0.3	-0.6	1.2
-1	0.15	-0.15	0.15
0	0.2	0	0
1	0.2	0.2	0.2
2	0.15	0.3	0.6
Total	1	-0.25	2.15

The expected value of X is given by

Variance of X is given by

Standard deviation of X is

c) Cumulative distribution function of X is

x	p	F(x)
-2	0.3	0.3
-1	0.15	0.45
0	0.2	0.65
1	0.2	0.85
2	0.15	1

d)

= 0.2 + 0.2 +0.15

= 0.55

e) For drawing histogram

Consider total frequency = 100

By using R

> x=seq(-2,2,1)
> y=c(30,15,20,20,15)
> f=rep(x,y)
> hist(f)
> t=seq(-2.5,2.5,1)
> hist(f,breaks=t,main="Histogram")