Question 1203397
The total probability must be 1.0

Add the probabilities and solve for n.

{{{P=1= 0.15 +0.20 +n +0.30+ 0.10}}}
{{{n=1-0.75=0.25}}}

n=0.25

Mean or average value, use n=0.25 value.

{{{E(x)=sum (xP(x))=0*0.15+  1*0.2+  2*0.25+  4*0.3+  5*0.1 =2.4}}}


Variance.

{{{Var(x)=sum((x-E(x))^2)P(x)=0.15(-2.4)^2+0.2(-1.4)^2+0.25(-.4)^2+0.3(1.6)^2+0.1(2.6)^2 =2.74}}}


B.A random variable X follows a Normal probability distribution and has a mean value of m=25 and a variance of Var=9. Calculate:

Standard deviation = {{{ s=sqrt(Var)=sqrt(9)=3}}}.

i.P(X < 21):

{{{z=(X-m)/s=(21-25)/3=-4/3}}}
{{{P(Z < -1.333)=0.0912}}}

ii.P(X > 26) 

{{{z=(X-m)/s=(26-25)/3=1/3}}}
{{{P(Z > 0.333)=1-P(Z < 0.333)=1-0.631=0.369}}}


iii.P( 23 < X < 28):


{{{a=(X-m)/s=(23-25)/3=-2/3}}}
{{{b=(28-25)/3=1}}}
{{{P(-0.667 <Z < 1)=P(Z < 1)-P(Z < -0.667)=0.841-0.252=0.589}}}