In this topic you turn a distribution into single numbers. Multiply each value by its probability and add. That gives the mean E[X]. Apply the same weights to g(x) to get E[g(X)]. Variance is the average squared distance from the mean. A problem belongs here when it gives a pmf or a density and asks for a mean, expected value, variance, standard deviation, or the average of something built from X. You will practice: weighted-average expectation, LOTUS, the variance shortcut, linear transforms.
A packet sent over a noisy network link needs X retransmissions before it gets through, where X has pmf P(X=0) = 0.4, P(X=1) = 0.3, P(X=2) = 0.2, P(X=3) = 0.1. Compute E[X].
Answer:
Apply the definition of expectation for a discrete random variable.
The voltage X across a 1-ohm resistor is uniformly distributed on [0, 2], so its density is f(x) = 1/2 for 0 ≤ x ≤ 2. The instantaneous power dissipated is W = X². Use LOTUS to compute E[W].
Answer:
By LOTUS there is no need to find the distribution of W. Integrate against the density of X.
A basketball player is sent to the free-throw line for two shots. Let X be the number of points she scores, with pmf P(X=0) = 0.1, P(X=1) = 0.3, P(X=2) = 0.6. Compute E[X].
Answer:
Expectation is the probability weighted average of the values.
A greenhouse controller logs the daily average temperature C in degrees Celsius, and long records show E[C] = 25. The display converts to Fahrenheit via F = 1.8·C + 32. What is E[F]?
Answer:
F is a linear transform of C, so use linearity of expectation. No new distribution is needed.
A subway busker counts the number X of listeners who drop a tip during one song. From experience, X has pmf P(X=0) = 0.2, P(X=1) = 0.5, P(X=2) = 0.3. Find E[X].
Answer:
Apply the definition of expectation as a probability weighted average.
A karaoke lounge charges a flat entry fee of 40 plus 5 per song sung, so a customer who sings X songs pays T = 40 + 5X. If a typical customer sings E[X] = 6 songs, find the expected total charge E[T].
Answer:
Use linearity of expectation for a linear transform.
A claw machine gives a prize on some plays. The prize value X in dollars is 0 with probability 0.8 and 5 with probability 0.2. Find E[X].
Answer:
Weight each value by its probability.
A gardener plants X seed trays in a morning, where E[X] = 5. Each tray holds 6 seedlings, and she always adds 2 extra seedlings in pots. The total number planted is N = 6X + 2. Find E[N].
Answer:
Expectation passes straight through aX + b.
At a gelato stand, a customer orders X scoops, where P(X=1) = 0.3, P(X=2) = 0.5, P(X=3) = 0.2. Find E[X].
Answer:
Weight each value by its probability and add.
On a random morning, the number X of elevators out of service in an office tower has pmf P(X=0) = 0.5, P(X=1) = 0.3, P(X=2) = 0.2. Find E[X²].
Answer:
Apply LOTUS: square each value but keep the same weights.
On each ferry crossing, X foot passengers arrive too late to board, where P(X=0) = 0.2, P(X=1) = 0.5, P(X=2) = 0.3. Find Var(X).
Answer:
Find both moments, then use the shortcut.
An editor counts X typos per chapter, and Var(X) = 4. A chapter's quality score is W = 10 − 3X. Find Var(W).
Answer:
The shift does nothing to variance and the coefficient comes out squared.
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