Random Variables and Mathematical Expectation MCQs 2026

A random variable is

1. Aa numerical function assigning a real number to each outcome in the sample space✓
2. Balways a constant✓
3. Conly a probability✓
4. Dthe sample space itself✓

A discrete random variable takes

1. Aonly every real number✓
2. Bonly zero✓
3. Cvalues with density only✓
4. Da countable set of values (finite or countably infinite)✓

A continuous random variable takes

1. Aonly integers✓
2. Bonly 0 and 1✓
3. Ca finite list always✓
4. Dvalues in an interval (uncountably many)✓

The probability mass function (PMF) p(x) gives

1. Adensity at x for continuous X✓
2. Bcumulative probability always✓
3. CP(X = x) for discrete X✓
4. Dvariance of X✓

A valid PMF must satisfy

1. Ap(x) can be negative✓
2. Bp(x) ≥ 0 and Σ p(x) = 1 over all x✓
3. Csum equals 0✓
4. Dp(x) unbounded above✓

The probability density function (PDF) f(x) for continuous X satisfies

1. Af(x) equals P(X=x)✓
2. Bf(x) ≥ 0 and ∫f(x)dx = 1 over all x✓
3. CΣf(x)=1✓
4. Df(x) can be negative✓

For continuous X, P(a ≤ X ≤ b) equals

1. Af(a) + f(b)✓
2. Bf(a) × f(b)✓
3. CΣ f(x)✓
4. D∫_a^b f(x) dx✓

The cumulative distribution function F(x) is defined as

1. AP(X = x) only✓
2. BP(X ≤ x)✓
3. CP(X ≥ x) always✓
4. DE(X)✓

For any CDF, lim_{x→−∞} F(x) equals

1. A0✓
2. B1✓
3. C0.5✓
4. Dundefined✓

For any CDF, lim_{x→+∞} F(x) equals

1. A1✓
2. B0✓
3. C0.5✓
4. Dinfinity✓

A CDF is always

1. Adecreasing✓
2. Balways linear✓
3. Cnegative✓
4. Dnon-decreasing and right-continuous✓

For discrete X, F(x) is

1. Aa step function with jumps at the values of X✓
2. Balways smooth✓
3. Calways zero✓
4. Dconstant✓

The expected value E(X) of discrete X equals

1. AΣ p(x) only✓
2. BΣ x only✓
3. CΣ x · p(x)✓
4. DΣ x² p(x)✓

For X with PMF: x=0,p=0.2; x=1,p=0.5; x=2,p=0.3, E(X) equals

1. A1.0✓
2. B1.1✓
3. C1.3✓
4. D0.8✓

For the same distribution, E(X²) equals

1. A1.7✓
2. B1.1✓
3. C1.21✓
4. D2.0✓

For the same distribution, Var(X) equals

1. A1.7✓
2. B0.6✓
3. C0.49✓
4. D1.1✓

Variance Var(X) is defined as

1. AE(X)² only✓
2. BE[(X − μ)²] = E(X²) − [E(X)]²✓
3. CE(X) only✓
4. Dstandard deviation squared always incorrectly as E(X)✓

The standard deviation σ equals

1. AVar(X) squared✓
2. Bthe positive square root of Var(X)✓
3. CE(X) − μ✓
4. DP(X ≤ μ)✓

Linearity of expectation gives E(aX + b) as

1. AaE(X) × b✓
2. BE(X) + a + b✓
3. Ca²E(X)✓
4. DaE(X) + b✓

If X and Y are independent, E(XY) equals

1. AE(X) + E(Y)✓
2. BE(X) / E(Y)✓
3. CE(X)E(Y)✓
4. DE(X²)✓

Var(aX + b) equals

1. Aa²Var(X)✓
2. BaVar(X) + b✓
3. CaVar(X)✓
4. DVar(X) + b²✓

If X and Y are independent, Var(X + Y) equals

1. AVar(X) × Var(Y)✓
2. BVar(X) + Var(Y)✓
3. CVar(X) − Var(Y)✓
4. D[Var(X) + Var(Y)]²✓

The PMF of a Bernoulli(p) trial is

1. AP(X=0)=p always✓
2. Buniform on 0,1,2✓
3. CP(X=1)=1 always✓
4. DP(X=1)=p, P(X=0)=1−p✓

For Bernoulli(p), E(X) equals

1. A1 − p✓
2. Bp(1 − p)✓
3. Cp✓
4. Dp²✓

For Bernoulli(p), Var(X) equals

1. Ap²✓
2. Bp(1 − p)✓
3. C1 − p✓
4. Dp✓

The mode of a distribution is

1. Athe value of x with highest probability (discrete) or highest density (continuous)✓
2. Balways the mean✓
3. Calways the median✓
4. DE(X²)✓

The median m satisfies

1. AP(X ≤ m) ≥ 0.5 and P(X ≥ m) ≥ 0.5✓
2. BP(X = m) = 1 always✓
3. Cm = E(X) always✓
4. Dm = mode always✓

The relationship between PDF and CDF is

1. AF(x) = f(x) × f(x)✓
2. Bf(x) = F(x) − 1✓
3. Cf(x) = dF(x)/dx where derivative exists✓
4. Dno relationship✓

For discrete X, recovering PMF from CDF: p(x) equals

1. AF(x) × 2✓
2. BF(x) − F(x−) (jump at x)✓
3. C1 − F(x) always✓
4. DΣ F(x)✓

Chebyshev's inequality bounds

1. AP(X = μ) exactly✓
2. BP(|X − μ| ≥ kσ) ≤ 1/k² for k > 0✓
3. Conly normal distributions✓
4. DVar(X) = 0✓

A fair die score X. E(X) equals

1. A3.5✓
2. B3✓
3. C4✓
4. D3.6✓

For fair die X, Var(X) equals

1. A35/12✓
2. B7/12✓
3. C2.5✓
4. D3.5✓

If Y = 2X + 3 and E(X)=4, Var(X)=5, then E(Y) equals

1. A8✓
2. B13✓
3. C11✓
4. D10✓

With the same Y = 2X + 3, Var(Y) equals

1. A10✓
2. B5✓
3. C25✓
4. D20✓

Joint PMF p(x,y) gives

1. AP(X = x and Y = y)✓
2. BP(X = x) only✓
3. CE(XY) always✓
4. DCov(X,Y)✓

Marginal PMF p_X(x) is found by

1. Asumming p(x,y) over all y✓
2. Bmultiplying over y✓
3. Csetting y = x always✓
4. Dintegrating CDF only✓

X and Y are independent iff

1. AE(X) = E(Y)✓
2. Bp(x,y) = p_X(x) p_Y(y) for all x,y✓
3. CVar(X) = Var(Y)✓
4. Dρ = 1✓

Cov(X,Y) = E(XY) − E(X)E(Y). If independent, Cov(X,Y) equals

1. A1✓
2. BVar(X)✓
3. C0✓
4. DE(X)E(Y)✓

Correlation ρ(X,Y) equals

1. ACov(X,Y)²✓
2. Bσ_X + σ_Y✓
3. CCov(X,Y) / (σ_X σ_Y)✓
4. DE(XY) only✓

A function g(X) is a random variable. E[g(X)] for discrete X equals

1. Ag(E(X)) always✓
2. BΣ g(x) p(x)✓
3. CΣ g(x) only without weights✓
4. DE(X)^g✓

Jensen's inequality for convex g states E[g(X)] is

1. Aalways equal to g(E(X))✓
2. B≥ g(E(X)) for convex g✓
3. C≤ g(E(X)) for convex g✓
4. Dunrelated to convexity✓

The moment generating function M(t) = E(e^{tX}) helps find moments because

1. AM(t) equals Var(X)✓
2. BM(t) is always 1✓
3. CE(X^n) equals the nth derivative of M(t) at t=0✓
4. Dderivatives give only median✓

Percentile p_k satisfies

1. AP(X = p_k) = k✓
2. Bp_k = E(X)✓
3. Cp_k = mode✓
4. DP(X ≤ p_k) = k/100 for continuous X✓

Interquartile range (IQR) equals

1. AQ1 + Q3✓
2. Bmedian − mean✓
3. Cσ²✓
4. DQ3 − Q1✓

For X uniform on {1,2,3,4}, E(X) equals

1. A2.5✓
2. B2✓
3. C3✓
4. D2.25✓

A RV with E(X)=μ and Var(X)=0 must satisfy

1. AP(X=0)=1✓
2. Buniform distribution✓
3. CP(X=μ)=0✓
4. DP(X = μ) = 1✓

The law of large numbers states that

1. Asample mean equals μ for any n✓
2. BVar(X) goes to infinity✓
3. Cthe sample mean converges to μ as n increases✓
4. DPMF changes with n✓

A probability histogram for discrete data uses

1. Asmooth curves only✓
2. Bnegative heights✓
3. Careas unrelated to probability✓
4. Dbar heights equal to probabilities (or relative frequencies)✓

The coefficient of variation CV equals

1. Aμ / σ✓
2. Bσ² / μ✓
3. Cμ × σ✓
4. Dσ / μ (standard deviation divided by mean)✓