GATE Data Science and Artificial Intelligence 2024 | Sample Paper | Question: 10

Question

$\mathrm{X}$ is a uniform distribution random variable with support in $[-2,2] \mathrm{U}[99.5$, $100.5]$. The mean of $\mathrm{X}$ is __________.

Answer 1

Ans: 20

 

Comments

@Mahanth Yalla $\frac{1}{5}x$ is an odd function, in the range $-2$ to $2$ the integral will come out to be zero.

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my bad, fooled by the diagram of the pdf. thanks for correcting me…

Answer 2

Since the PDF is a constant c within the support and 0 outside, we can split the integral into two parts:

$E(X) = ∫_{-2}^{2} x * c dx + ∫_{99.5}^{100.5} x * c dx$

Now, we can factor out the constant c and evaluate the integrals:

E(X) = c * [x^2/2]_{-2}^{2} + c * [x^2/2]_{99.5}^{100.5}

E(X) = c * (4/2 - 4/2) + c * (10100.25/2 - 9900.25/2)

E(X) = c * (100)

To find the value of c, we use the fact that the total probability over the support must equal 1:

1 = ∫_{-2}^{2} c dx + ∫_{99.5}^{100.5} c dx

1 = c * (4 + 1)

c = 1/5

Therefore, the expected value of X is:

E(X) = (1/5) * (100)

E(X) = 20

So the answer is 20.