Answer and Explanation:
Given : The random variable x has the following probability distribution.
To find : 
a. Is this probability distribution valid? Explain and list the requirements for a valid probability distribution.
b. Calculate the expected value of x. 
c. Calculate the variance of x. 
d. Calculate the standard deviation of x.
Solution : 
First we create the table as per requirements,
x      P(x)         xP(x)           x²            x²P(x)
0    0.25           0               0                0
1     0.20        0.20             1              0.20
2    0.15          0.3               4             0.6
3    0.30         0.9               9             2.7
4    0.10          0.4               16             1.6
    ∑P(x)=1     ∑xP(x)=1.8               ∑x²P(x)=5.1
a) To determine that table shows a probability distribution we add up all five probabilities if the sum is 1 then it is a valid distribution.


Yes it is a probability distribution.
b) The expected value of x is defined as

c) The variance of x is defined as 

d) The standard deviation of x is  defined as 


