Question

The brightness level Y of a pixel of a TV screen is 90% of the time described by a Gaussian random variable with mean 128 and variance 64. However, 5% of the time the pixel is defective and it is never on (always black), so that Y = 0. The remaining 5% of the time the pixel is always white, so Y = 255. For this mixed random variable 1. Give the mathematical form of the pdf $f_Y(y)$ using delta functions $\delta(.)$ and gaussian pdf. Sketch the pdf. 2. Give the mathematical form of the cdf $F_Y(y)$ using unit step function $u(.)$ and sketch the pdf. 3. What is the probability the pixel brightness $Y \ge 200$?

          The brightness level Y of a pixel of a TV screen is 90% of the time described by a Gaussian random variable with mean 128 and variance 64. However, 5% of the time the pixel is defective and it is never on (always black), so that Y = 0. The remaining 5% of the time the pixel is always white, so Y = 255. For this mixed random variable
1. Give the mathematical form of the pdf $f_Y(y)$ using delta functions $\delta(.)$ and gaussian pdf. Sketch the pdf.
2. Give the mathematical form of the cdf $F_Y(y)$ using unit step function $u(.)$ and sketch the pdf.
3. What is the probability the pixel brightness $Y \ge 200$?

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The brightness level Y of a pixel of a TV screen is 90% of the time described by a Gaussian random variable with mean 128 and variance 64. However, 5% of the time the pixel is defective and it is never on (always black), so that Y = 0. The remaining 5% of the time the pixel is always white, so Y = 255. For this mixed random variable
1. Give the mathematical form of the pdf fY(y) using delta functions δ(.) and gaussian pdf. Sketch the pdf.
2. Give the mathematical form of the cdf FY(y) using unit step function u(.) and sketch the pdf.
3. What is the probability the pixel brightness Y ≥ 200?

Added by Elena C.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

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Solved by Expert Kirsty Gledhill

Step 1

The mathematical form of the pdf fáµ£(y) can be expressed as: fáµ£(y) = 0.05 * Î´(y) + 0.05 * Î´(y - 255) + 0.9 * (1 / (âˆš(2Ï€) * Ïƒ)) * exp(-(y - Î¼)^2 / (2 * Ïƒ^2)) where Î´(y) represents the delta function, Î´(y - 255) represents the delta function shifted Show more…

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The brightness level Y of a pixel of a TV screen is 90% of the time described by a Gaussian random variable with mean 128 and variance 64. However, 5% of the time the pixel is defective and it is never on (always black), so that Y = 0. The remaining 5% of the time the pixel is always white, so Y = 255. For this mixed random variable: 1. Give the mathematical form of the pdf fáµ£(y) using delta functions (Î´) and Gaussian pdf. Sketch the pdf. 2. Give the mathematical form of the cdf Fáµ§(y) using the unit step function u(.) and sketch the pdf. 3. What is the probability the pixel brightness Y > 200?

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