P(H)+P(T)=1
       P(H)=3P(T)

 3P(T)+P(T)=1
      4P(T)=1
       P(T)=1/4
       P(H)=3P(T)
P(H)+3(1/4)=3/4

This is a binomial distribution:

Let x be the number of heads:

x                                     P(x)
0    C(5,0)(3/4)^0(1/4)^5 =  1*3^0/4^5 =    1*1/1024 =   1/1024                     
1    C(5,1)(3/4)^1(1/4)^4 =  5*3^1/4^5 =    5*3/1024 =  15/1024
2    C(5,2)(3/4)^2(1/4)^3 = 10*3^2/4^5 =   10*9/1024 =  90/1024
3    C(5,3)(3/4)^3(1/4)^2 = 10*3^3/4^5 =  10*27/1024 = 270/1024
4    C(5,4)(3/4)^4(1/4)^1 =  5*3^4/4^5 =   5*81/1024 = 405/1024
5    C(5,5)(3/4)^5(1/4)^0 =  1*3^5/4^5 =  1*243/1024 = 243/1024   

mean = n*p = 5*(3/4) = 15/4 = 3.75
variance = n*p*(1-p) = 5*(1/4)(3/4) = 15/16 = 0.9375

Edwin