Question Details

If probability distribution is given by,

x 0 1 2 3
p(x) 8 a 1 30 4 a 1 30 2 a + 1 30 b
If it is given that  σ 2 + μ 2 = 2  where  σ  is standard deviation and  μ  is mean of distribution then  a b  is

Options

A

2271

B

11071

C

22071

D

111071

Show Answer

Correct Answer :

Option D

111071

Solution :

The correct option is 111071.

To find the value of ab, we utilize two key properties of probability distributions.

Step 1: Use the property that the total probability must equal 1.
The sum of all probabilities in a discrete probability distribution is always equal to 1:

p(x)=p(0)+p(1)+p(2)+p(3)=1

Substitute the given values of p(x) into the equation:

8a-130+4a-130+2a+130+b=1

Combine the fractions with the common denominator of 30:

(8a-1)+(4a-1)+(2a+1)30+b=1

Simplify the numerator:

14a-130+b=1

Multiply the entire equation by 30 to clear the fraction:

14a-1+30b=30

14a+30b=31

(Equation 1)

Step 2: Use the formula for variance and mean.
We are given that σ2+μ2=2.
Recall that the variance σ2 is defined as:

σ2=E[X2]-μ2

Rearranging this gives:

E[X2]=σ2+μ2=2

Now, calculate E[X2] using E[X2]=x2·p(x):

E[X2]=02·8a-130+12·4a-130+22·2a+130+32·b

E[X2]=0+4a-130+4(2a+1)30+9b

E[X2]=4a-1+8a+430+9b=12a+330+9b

Set E[X2]=2:

12a+330+9b=2

Multiply the entire equation by 30:

12a+3+270b=60

12a+270b=57

Divide the entire equation by 3:

4a+90b=19

(Equation 2)

Step 3: Solve the system of linear equations for a and b.
Multiply Equation 1 by 3 to match the coefficient of b in Equation 2:

3·(14a+30b)=3·31

42a+90b=93

Subtract Equation 2 (4a+90b=19) from this equation:

(42a+90b)-(4a+90b)=93-19

38a=74

a=7438=3719

Substitute a=3719 back into Equation 2 to find b:

43719+90b=19

14819+90b=19

90b=19-14819=361-14819=21319

b=21319·90=71570

Step 4: Compute the ratio ab.

ab=371971570=3719·57071

Since 57019=30:

ab=37·3071=111071

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